"9-NRLF 


572   273 


WALLS 


7 

LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

l^ceived        j^)   12    1893    .189 
Accessions  No.l\C(C(C\L^    .  Class  No, 

H  *  - 


RETAINING- WALLS  FOR  EARTH. 


THE  THEORY  OF  EARTH-PRESSURE 

AS  DEVELOPED   FROM  THE 

ELLIPSE  OF  STRESS. 


AN  APPENDIX  PRESENTING  THE   THEORY  OF 
PROF.   WEYRAUCH. 


BY 

MALVERD  A.  HOWE,  O.E., 

Professor  of  Civil  Engineering,  Rose  Polytechnic  Institute. 


Seconlr  Btiition,  3acbtscti  aitlr 


NEW  YORK: 

JOHN     WILEY    &    SONS, 
53  EAST  TENTH  STREET. 

1891, 
*Wx^ 

THE 

'UHIVERSITTl 


COPYRIGHT,  1891, 

BY 
JOHN  WILEY  &  SONS. 


FERRIS  BROS., 
ROBERT  DRCMMOND,  printers, 

Electrotype,;  pearl  street| 

414  &  446  Pearl  Street,  N 

Mew  York. 


CONTENTS. 


PART  I. 

PAGE 

NOMENCLATURE, vii 

FORMULAS  FOR  THE  THRUST  OF  EARTH, 1 

FORMULAS  FOR  THE  BREADTH  OF  BASE  OF  A  WALL,       ....  6 

FORMULAS  FOR  THE  DEPTH  OF  FOUNDATIONS 9 

EXAMPLES, 11 

PART  II. 

DEMONSTRATION  OF  THE  FORMULAS  FOR  THE  THRUST  OF  EARTH,     .       .  27 
DEMONSTRATION  OF  THE  FORMULAS  FOR  THE  BREADTH  OF  THE  BASE  OF 

A  WALL, 48 

DEMONSTRATION  OF  THE  Fo  MULAS  FOR  THE  DEPTH  OF  FOUNDATIONS,  54 


APPENDIX. 

WEYRAUCH'S  THEORY  OF  EARTH-PRESSURE,           ......  59 

REFERENCES, 103 

DIAGRAM  I,           10? 

TABLES,     ....,,.,...,,.  109, 


PREFACE. 


THE  first  edition  of  this  work  was  based  upon  the 
theory  advanced  by  Prof.  Weyrauch  in  1878,  but  owing 
to  the  length  of  the  demonstrations  used  by  him,  it 
was  thought  advisable  to  present  different  and  shorter 
demonstrations  in  this  edition.  To  show  that  the  new 
demonstrations  give  identical  results  with  those  obtained 
by  Prof.  Weyrauch,  his  demonstrations  have  been  given  in 
an  appendix  as  they  appeared  in  the  first  edition. 

The  new  demonstrations  are  based  upon  the  theory  first 
advanced  by  Prof.  Rankine  in  1858.  Those  readers  who 
are  familiar  with  Rankine's  Ellipse  of  Stress  can  omit 
pages  27  to  35,  inclusive,  in  following  the  demonstrations. 

An  attempt  has  been  made  to  present  the  theory  in  a 
shape  easily  followed  by  those  who  have  only  a  knowledge 
of  algebra,  geometry,  and  trigonometry;  whenever  cal- 
culus has  been  resorted  to,  the  work  has  been  simplified  as 
much  as  possible.  For  convenience  in  practice,  the  formu- 
las have  been  arranged  in  a  condensed  shape  in  Part  I, 
and  are  followed  by  numerous  examples  illustrating  their 
application. 

The  values  of  various  coefficients  have  been  computed 
and  tabulated  and  will  be  found  to  very  materially  decrease 
the  labor  of  substitution  in  the  formulas, 

v 


vi  PREFACE. 

It  is  hoped  that  the  introduction  of  a  brief  treatment  of 
the  supporting  power  of  earth  in  the  case  of  foundations, 
as  well  as  the  formula  for  determining  the  breadth  of  the 
base  of  a  retaining- wall,  will  prove  acceptable. 

For  valuable  help  in  the  verification  of  proofs  of  formu- 
las, and  the  critical  reading  of  the  whole  text,  I  acknowl 
edge  the  kind  assistance  of  Prof.  Thos.  Gray. 

M.  A.  H. 

TEBRE  HAUTE,  IND.  ,  March,  1891. 


NOMENCLATURE. 


0  =  the  angle  of  repose,  or  the  maximum  angle  which 
any  force  acting  upon  any  plane  within  the  mass 
of  earth  can  make  with  the  normal  to  the  plane. 
e  =  the  angle  made  by  the  surface  of  the  earth  with  the 
horizontal;  e  is  positive  when  measured  above  and 
negative  when  measured  below  the  horizontal. 
a  =  the  angle  which  the  back  of  the  wall  makes  with 
the  vertical  passing  through  the  heel  of  the  wall; 
a  is  positive  when  measured  on  the  left  and  nega- 
tive when  measured  on  the  right  of  the  vertical. 
6  =  the  angle  which  the  direction  of  the  resultant  earth- 
pressure  makes  with  the  horizontal. 

0'  =  the  angle  of  friction  between  the  wall  and  its  foun- 
dation. 
0"  =  the  angle  of  friction  between  the  back  of  the  wall 

and  the  earth. 

If  =  the  vertical  height  of  the  wall  in  feet. 
h  '=  the  depth  of  earth  in  feet  which  is  equivalent  to  a 

given  load  placed  upon  the  surface  of  the  earth. 
B'  =  the  width  in  feet  of  the  top  of  the  wall. 
B  =  the  width  in  feet  of  the  base  of  the  wall. 
Q  =  the  distance  in  feet  from  the  toe  of  the  wall  to  the 
point  where  R  cuts  the  base. 

vii 


Vlll  NOMENCLATURE. 

P  —  the  resultant  earth-pressure  in  pounds  against  a  ver- 
tical wall. 

E  =  the  resultant  earth-pressure  in  pounds  against  any 
wall. 

R  =  the  resultant  pressure  in  pounds  on  the  base  of  the 
wall. 

G  =  the  total  weight  in  pounds  of  material  in  the  wall. 

y  =  the  weight  in  pounds  of  a  cubic  foot  of  earth. 

W  =  the  weight  in  pounds  of  a  cubic  foot  of  wall. 

p  =  the  intensity  of  the  pressure  in  pounds  on  the  base 

of  the  wall  at  the  toe. 
p'  —  the  intensity  of  the  pressure  in  pounds  on  the  base 

of  the  wall  at  the  heel. 

p9  =  the  average  intensity  of  the  pressure  in  pounds  on 
the  base  of  the  wall, 

x  —  H  tan  a. 


OF  THE 

UNIVERSITY 


RETAINING-WALLS  FOR  EARTH 


FORMULAS   FOR  EARTH-PRESSURE. 

IN  the  following  formulas  a  and  e  are  considered  as 
positive,  and  the  wall  is  assumed-to  be  one  foot  long. 

CASE  I.  General  case  of  inclined  earth-surface  and  in- 
clined back  of  wall. 


= 

%    cos2  a  cos  e 


t   / 

\/ 
V 


.  (  cos  e  —  i/cos'2  e  —  cos'2  0  )  a 

in2  a  -{-  cos9  (e  —  a)  1 

(  cos  e  -f-  I'cos*  e  —  cos2  <p  ) 


.    j  cos  e  —  I/cos2  e  —  cos*  <f>  \ 
-f  2  sin  esiu  acos  (e  -  a)  4  -r  -  _=  - 

(  cos  e  +  I/cos2  e  —  cos^  </>  ) 


'  ^    ' 


or 


V(C) 


tan  *  : 


sn  tf  cos 


e  -f  sin  e  cos  (e  —  oi]A          ,    A 

- 


or 


tan  d  = 


-  - 

cos  (e  —  o-) 


2  RETAIN1NG-WALLS  FOR  EARTH. 

where 


cos  e  —  l/cos2  e  —  cos"  0  ,  7. 

A  =  cos  e  —  — =  == .  .     .      (a) 

cos  e  -f  r  cos2  e  —  cos2  <p 

CASE  II.  Surface  of  earth  inclined  and  a  =  0. 


*    (  cos  e  -f  *  cos"  e  —  cos2  0 

From  Diagram  I  the  values  of  A  can  be  found  for  all 
values  of  cf>  from  0°  to  90°  and  of  e  from  0°  to  90°,  vary- 
ing by  5°. 


or  for  all  vertical  walls  tlie  direction  of  the  earth-pressure 
is  parallel  to  the  surface  of  the  earth. 

CASE  III.   The  surface  of  the  earth  parallel  to  the  surface 
of  repose. 


E  _  H^r  cos  (0  —  ^)  |/sinj  a  -f-  cosa  (0  -  a)  ,^\ 

2    cos2  a  cos  0  ^   -j-  2  sin  <^  sin  0  cos  (0  —  fl')' 

tan  <J  =  «i""  +  ri"_0^I0JL£).  .  (3f,) 

COS  0  COS   (0  —  tf) 

CASE  IV.   77/e  surface  of  Ike  earth  parallel  to  the  surface 
of  repose  and  the  back  of  the  wall  vertical. 

e  =  0     and     a  =  0. 

/TV 
^=   -^-cos0 (4) 

<y  =  0 (4« 


FORMULAS  FOR  EARTH-PRESSURE. 

CASE  V.  The  surface  of the  ear  Hi  horizontal 


=  ^Z  |/tan2  a  +  tan1  (45°  -  |Y       .       (5) 

/v  \  / 


tan 


tan'  (45°.- 


CASE  VI.    7%c  surface  of  the  earth  horizontal  and  the 
hack  of  the  wall  vertical. 

e  =  0    and     a  =  0. 

--     ....    (6) 


CASE  VII.  Fluid  pressure. 

e  =  0  =  0. 

*=0^  .......       (7) 

2  cos  tf 

<^  =  a  .........     (7/0 

GRAPHICAL  CONSTRUCTIONS  FOR  DETERMINING  THE 
THRUST  OF  EARTH. 

The  following  constructions  are  perfectly  general,  and 
apply  to  any  plane  within  a  mass  of  earth.     When  applied 


4  .         RETAINING-WALLS  FOU  EAHTti. 

for  determining  the  thrust  of  earth  against  a  retaining-wall, 
a  and  e  are  taken  as  positive. 

*  Construction  (a). 

Let  BE  represent  the  surface  of  the  earth  and  BA  the 
back  of  the  wall.  Draw  AF  parallel  to  BE,  and  at  any 
point  D  in  AF  lay  off  DF  equal  to  the  vertical  DE.  Draw 


FIG.  1. 

FG  horizontal,  and  FH,  making  the  angle  0  with  DF. 
With  any  point  /in  DF  describe  the  arc  KI  tangent  to 
HF  at  /  cutting  FG  at  /i,  and  draw  GL  parallel  to  KJ\ 
with  L  as  a  centre  and  LF  as  radius,  describe  the  circum- 
ference FQON  cutting  AD  at  N.  Through  TV  draw  NO 

*  See  "Theorie  desErddruckes  auf  Gnmd  der  neuercn  Anschau- 
ungen,"  by  Prof.  Weynmch,  1881. 


FORMULAS  FOR  EARTH  PRESSURE. 


parallel  to  AB  cutting  the  circumference  FQON  at  0\ 
at  A  draw  A  0  equal  to  OG  and  normal  to  AB\  the  area 
of  the  triangle  ABC  multiplied  by  y  will  be  the  thrust  of 
the  earth  on  the  wall. 

To  determine  the  direction  of  the  thrust  E,  prolong  OG 
to  Q\  then  QN  will  be  the  direction  of  the  thrust. 

This  thrust  acts  on  the  wall  at  \AB  below  B. 

*  Construction  (b). 

Let  BQ  represent  the  surface  of  the  earth,  and  BA  the 
back  of  the  wall.  Draw  AD  parallel  to  B Q,  and  at  any 


FIG.  2. 


point  D  in  AD  draw  the  vertical  DG  equal  to  the  normal 
draw  DM  making  the  angle  0  with  the  normal  DQ. 


*  This  construction  follows  directly  from  Rankine's  Ellipse  of 
Stress,     §ee  Raukiiie's  Applied  Mechanics. 


6  RETA1N1NG-WALLS  FOR  EARTH. 

At  any  point  J  in  DQ  as  a  centre,  describe  the  arc  IK  tan- 
gent to  DM  cutting  DG  at  K,  and  draw  OL  parallel 
to  JK.  Bisect  the  angle  QLG,  and  at  A  draw  A  P  parallel 
to  LR.  At  A  draw  AN  normal  to  AB  and  equal  to  DL; 
with  JV  as  a  centre  and  AN  as  radius,  describe  an  arc 
AP  cutting  AP  at  P]  connect  P  and  N9  and  make  NO 
equal  to  LG\  with  ^4  as  a  centre  and  J  0  as  a  radius,  de- 
scribe the  arc  00  cutting  AN  at  (7;  then  the  area  of  the 
triangle  ABC  multiplied  by  y  will  be  the  thrust  against 
the  wall.  The  direction  of  this  thrust  is  parallel  to  A  0 
and  it  is  applied  at  \AB  below  B. 

The  constructions  (a)  and  (b)  give  identical  results  in 
every  case. 

TKAPEZOIDAL  AND  TKIAKGULAR  WALLS. 

Formulas  for  the  width  of  the  base  of  trapezoidal  walls 
under  the  condition  that  the  resultant  R  cuts  the  base  at 
a  point  distant  from  the  toe  of  the  wall  equal  to  one  third 
the  width  of  the  base,  or  Q  =  ^B. 

CASE  I.  The  general  case  in  which  the  back  of  the  wall 
is  inclined,  and  E  makes  an  angle  ivith  the  horizontal. 


H  cos  6  +  x  sin  d    +  2£'x  +B'\  .     (8) 


CASE  II.   The  lack  of  the  ivall  vertical. 


t\T?  \  9  Ti] 

j±  sin  S  +  £')  =  ~w  cos  d  +  B'\        (9) 


FORMULAS  FOR  EARTH- PRESSURE.  " 

CASE  III.   The  lack  of  the  wall  vertical  and  the  thrust 
normal  to  the  wall. 


(10) 


If  B  —  B'  and  x  —  0,  the  section  of  the  wall  is  a  rec 
tangle,  and  (9)  becomes 


±E  %E 

B  -FTT7>  sin  o  =  -yy  cos  0, 


and  (10)  becomes 


8  RETAINING  -WALLS  FOR  EARTH. 

Formulas  for  the  width  of  the  base  of  triangular  walls 
under  the  condition  that  the  resultant  R  cuts  the  base  at 
a  point  distant  from  the  toe  of  the  wall  equal  to  one  third 
the  width  of  the  base,  or  Q  =  ^B. 

CASE  I.  The  general  case  in  which  the  hack  of  the  wall 
is  inclined)  and  E  makes  an  angle  with  the  horizontal. 


B*+B         psin  tf  -  *   =          (tfcos  tf  +  a  sin  #).     (11) 

CASE  II.   The  hack  of  the  wall  vertical. 


CASE  III.  The  hack  of  the  wall  vertical,  and  the  thrust 
normal  to  the  wall. 

x  =  0     and     d  =  0. 

*=*/W-    ......     (13) 

TJie  above  formulas  do  not  contain  the  condition  that  R 
shall  not  make  an  angle  greater  than  <f>'  with  the  normal  to 
the  base  of  the  wall. 

From  Fig.  3, 


which  expresses  the  condition  under  which  the  wall  will 
not  slidct 


FORMULAS  FOR  EARTH-PRESSURE.  9 

DEPTH  OF  FOUNDATIONS. 

CASE  I.  When  the  intensity  of  the  pressure  on  the  earth 
is  uniform. 

Letting  x'  equal  the  depth  of  the  foundation  below  the 
surface, 

<r'-  ;jn(l  -  sin  0)* 

(1  +  sin0)>-  JF(L-sin  0)2' 

when  the  weight  of  the  foundation  is  included;  and 


sm 


y 


when  the  weight  of  the  foundation  is  not  included. 

x'  is  the  minimum  depth  to  which  the  foundation  must 
be  extended  for  equilibrium.  The  actual  depth  should  be 
based  upon  the  minimum  value  which  0  is  likely  to  have, 
under  any  condition  of  the  earth. 

CASE  II.  When  the  intensity  of  the  pressure  on  the  earth 
is  uniformly  varying. 

x,  _  P.  0  ~  sin  0)8 
~  y    1  +  sin'~0  '  ' 

where  x'  is  the  minimum  depth  to  which  the  foundation 
must  be  extended  for  equilibrium; 

_  1        sin  0 
^  ~  3  1  -f  sin8  0'     ..... 

where  x0  is  the  maximum  distance  from  the  centre  of  the 
base  of  the  foundation  to  the  point  where  the  resultant 
pressure  cuts  the  base  of  the  foundation. 


10  RETAINING-WALLS  FOR  EARTH. 

ABUTTING  POWER  OF  EAKTH. 

_  (x'Y  Y  1  +  sin  0  /91  x 

2       1  -  sin  0' 

where  P  represents  the  maximum  resultant  pressure  which 
horizontal  earth  can  resist,  when  P  is  applied  against  \\ 
vertical  plane  of  the  depth  x'. 

APPLICATIONS. 

The  determination  of  the  earth-pressure  by  the  pre- 
ceding formulas  and  graphical  constructions  is  a  very 
simple  operation  when  the  angle  0  has  been  determined  or 
assumed.  That  care  and  judgment  be  used  in  assuming 
the  value  of  0  is  very  important,  since  a  change  of  a  few 
degrees  in  the  value  of  0  sometimes  causes  a  large  change 
in  the  value  of  E.  An  inspection  of  Diagram  I  shows  that 
the  value  of  the  coefficient  A  increases  very  rapidly  as  0 
decreases. 

When  the  earth  to  be  retained  contains  springs,  the 
bank  must  be  thoroughly  drained  if  it  is  to  be  retained  by 
an  economical  tight  wall;  if  it  is  not  drained,  the  angle  0 
will  be  likely  to  become  very  small  as  the  earth  becomes 
wet. 

When  the  location  of  the  earth  to  be  retained  is  sub- 
jected to  jars,  the  value  of  0  will  be  decreased. 

Hence,  in  assuming  the  value  of  0,  the  engineer  must  be 
sure  that  the  value  assumed  will  be  the  least  value  which, 
in  his  judgment,  it  is  likely  to  have. 

In  constructing  the  Avail  the  judgment  and  authority  of 
the  engineer  must  again  be  exercised  in  order  that  the  wall 
be  constructed  as  designed. 

In  all  cases,  to  insure  perfect  drainage  between  the  back 


FORMULAS  FOR  EARTH-PRESSURE.  11 

of  the  wall  and  the  earth,  numerous  "  weep-holes"  should 
be  provided  in  the  body  of  the  wall,  or  proper  arrange- 
ments made  to  carry  away  the  water  at  the  base  of  the  wall. 
To  facilitate  drainage,  the  backing  resting  against  the  wall 
sfionld  be  sand  or  gravel. 

In  no  case  should  water  be  permitted  to  get  under  the 
foundation  of  the  wall,  neither  should  the  earth  in  front 
of  the  wall  be  allowed  to  become  wet. 

In  cold  localities  the  back  of  the  wall  near  the  top  should 
have  a  large  batter  to  prevent  the  frost  from  moving  the 
top  courses  of  stone.  As  a  guard  against  sliding,  the 
courses  of  the  wall  should  have  very  rough  beds.  The 
strength  of  a  wall  is  increased  the  nearer  it  approaches  a 
monolith. 

Care  should  be  taken  to  have  the  foundation  broad  and 
deep  enough  to  prevent  sliding  and  upheaving  of  the  earth 
in  front.  In  clay  the  foundation  should  be  deep,  while  in 
sand  or  gravel  it  may  be  broad  and  shallow. 

The  following  examples  illustrate  the  application  of  the 
formulas : 

Ex.  1.  Design  a  trapezoidal  wall  of  sandstone,  weighing 
150  Ibs.  per  cubic  foot,  having  a  width  of  3  ft.  on  top,  a 
height  of  30  ft.,  and  the  back  inclining  forward  5°,  to  re- 
tain a  bank  of  sand  sloping  upward  at  an  angle  of  20°. 

Data. 

y  -  100  Ibs.,  W=  150  Ibs.;  e  =  20°,  0  =  39°,  a  =  5°; 
H  =  30  ft.,  B'  =  3  ft.,  x  -  2.63  ft. 

1°.   Graphical  determination  of  the  values  of  E  and  6\ 
The  graphical  solution  of  the  problem  is  shown  in  Fig.  4, 

where  E  is  found  to  equal  15,000  pounds.     6  lies  between 

35°  and  3G°. 


12  RETA1NING-WALLS  FOR  EARTH. 

2°.  Algebraic  determination  of  E  and  d. 


=-      (B)  V(0) 


*  +  (E)A. 


FIG.  4. 


Substituting  the  values  of  B,  C,  D,  and  E  as  given  in  the 
tables,  and  that  of  A  as  given  by  Diagram  I,  this  becomes 


l/(0.008)+(1.057)(0264)a+(0.061)0.264, 
^-45,000  (1.036)  1/0098  =  14,500  Ibs. 


tan  d  = 


sin  a 


cos  (e  —  a)A 


+  tan  e,      .     .     (I' a) 


FORMULAS  FOR  EARTH-  PRESSURE.  13 


tan  6  =  0.705  =  tan  35°  11',  about. 

3°.  Algebraic  determination  of  the  value  of  B  under  the 
assumption  that  Q  =  %B. 


".     .     (8) 


B*  -j-  7.797?  =  172.53, 

B-  -  3.89  ±  A/172.53  -f  3^9a; 
.-.  B  =  13.69  —  3.89  =  9.80  ft; 

or,  practically,  10  feet  is  the  required  width  of  the  base. 

4°.  To  determine  if  the  watt  will  slide  on  a  foundation  of 
sandstone. 

From  (14), 

,  E  cos  d 


Taking  B  =  10  ft.,    (?  =      --  30x150  =  29250  Ibs. 

fl 


14  RETAINING-WALL8  FOR  EARTH. 

d  =  35°  11',  cos  d  =  0.817,  and  sin  6  =  0.576,  then 

Ecos  $  14500  X  0.817 

G  -f  E  sin  d  ~  29250  +  14500"x"oT57G  ~ 

From  Table  II,  the  value  of  tan  0'  for  masonry  is  0.6  to 
0.7;  hence  there  is  no  danger  of  the  wall  sliding  on  the 
foundation. 

5°.  To  determine  the  minimum  depth  to  which  the  foun- 
dation must  extend  consistent  with  the  stability  of  the  earth. 

First  determine  the  maximum  value  of  x0.     From  (20), 

1       sin  0 


3  1  +  sin3  0' 

where  0  must  be  assumed  at  its  minimum  value.     Assume 
that  the  minimum  value  of  0  in  this  case  is  30°;  then 


showing  that  the  resultant  must  cut  the  base  of  the  foun- 
dation within  0.133  feet  of  its  centre.  The  resultant  cuts 
the  base  of  the  wall  1.67  feet  from  the  centre- of  its  base; 
hence  the  width  of  the  foundation  must  be  increased. 

Assuming  that  the  depth  to  which  the  foundation  ex- 
tends is  4  feet,  and  that  it  is  vertical  in  the  rear;  then  the 
direction  of  the  resultant  pressure  (not  including  the  addi- 
tional weight  of  the  foundation)  will  cut  the  base  of  the 
foundation  7.93  feet  from  the  ret;r  or  heel.  The  required 
width  of  the  base  of  the  foundation  is  (7.93  —  0.13)2  = 
15.6;  say,  16  feet. 

The  value  of  ;?0  can  now  be  found,  which  corresponds 
to  the  assumed  value  of  x9  —  4  feet. 


FORMULAS  FOR  EARTH-PRESSURE.  15 

From  (19), 

,     1  +  sin*  0 
Po  ~  X  r(l  -  sin  0)2  > 

p9  =  400  '-^  =  2960  Ibs. 

The  average  intensity  of  the  pressure  on  the  base  of  the 
foundation  due  to  the  resultant  R  is 

29250  +  14500  sin  d 
16 

The  foundation  adds  an  intensity  equal  to  4  X  150  =  600 
pounds  approximately;  hence  the  actual  value  of  /?„  =  2350 
4.  600  =  2950  pounds ;  therefore,  if  tlie  foundation  has  a 
depth  of  4  feet  and  a  base  of  16  feet,  the  wall  will  not  sink 
nor  the  earth  in  front  of  the  wall  heave,  until  0  becomes 
less  than  30°. 

6°.  To  determine  if  the  wall  and  foundation  will  slide  on 
the  earth. 

This  is  resisted  in  two  ways — by  the  friction  between  the 
masonry  und  the  earth,  and  by  a  prism  of  earth  in  front  of 
the  wall. 

The  horizontal  force  tending  to  make  the  wall  slide 
equals  E  sin  6,  or  14500-0.576  =  8352  pounds.  The  hori- 
zontal force  tending  to  make  the  foundation  slide  equals 
the  resultant  earth-pressure  on  the  rear  face  of  the  founds* 
tion,  which  is  vertical  and  4  feet  in  height.  From  (6), 

E  =  j  (30+ J>!  _  ?C 
(         *  * 

or  E  =  12800  X  0.226  =  2893. 


16  RETAINING- WALLS  FOR  EARTH. 

Then  the  total  horizontal  force  tending  to  make  the  wall 
slide  is 

8352  +  2893  =  11245  Ibs. 

From  Table  II  the  tangent  of  the  angle  of  friction  be- 
tween masonry  and  moist  clay  is  0.33,  which  evidently  is 
much  smaller  than  the  tangent  of  the  actual  angle  of  fric- 
tion between  masonry  and  dry  earth.  Assume  this  tangent 
to  be  0.500. 

The  total  vertical  pressure  upon  the  base  of  the  foun- 
dation is  37600  pounds,  hence  the  ability  to  resist 
sliding  is  37600  (0.5)  =  18800  pounds,  which  is  much 
Lirger  than  11245;  hence  there  is  no  danger  of  the  wall 
slipping,  even  if  the  earth  in  front  of  the  wall  does  not  act. 

Ex.  2.  Design  a  trapezoidal  wall  of  sandstone  weighing 
150  Ibs.  per  cubic  foot,  having  a  width  of  3  ft.  on  top,  a 
height  of  30  ft.,  and  the  back  inclining  backward  15°,  to 
retain  a  bank  of  sand  sloping  upward  at  an  angle  of  30°. 

Data. 

y  =  100  Ibs.,  W  =  1501bs.;e  =  30°,  0  =  33°,  a  =-  15°; 
H  =  30  ft.,  B'  =  3  ft.,  x  =  8  ft. 

1°.   Graphical  determination  of  the  values  of  E  and  d. 

In  Fig.  5,  let  EG  represent  the  surface  of  the  earth,  and 
AB  the  back  of  the  wall.  Draw  AF  parallel  to  BG,  and 
from  any  point  D'  in  A  F  lay  off  D' F  equal  to  the  vertical 
D'G,  and  draw  FL  horizontal;  lay  off  the  angle  1FD'  —  <p 
—  33°,  and  locate  the  point  M  in  D'Fso  that  if  an  arc  be 
described  with  M  as  a  centre  and  LM  as  a  radius  the  arc 
will  be  tangent  to  IF\  then  with  M  as  a  centre  and  MF  as 
a  radius,  describe  the  circumference  FHJ  and  draw  JH 


FORMULAS  FOR  EARTH-PRESSURE. 


17 


parallel  to  AB\  at  A  draw  AL  perpendicular  to  AB  and 
equal  to  HI.     Then 


=       .JK 


To  determine  6",  prolong  HI  to  A"  and  draw  AV.  Then 
the  angle  which  tliis  line  makes  with  the  horizontal  is 
equal  to  d,  which  is  6°  to  7°  in  this  case. 


FIG.  5. 


2°.  Algebraic  determination  of  E  and  d. 

Substituting  in  (1)  and  remembering  that  a  is  negative, 


£  =  45000  (0,875)  /O.OG7  -j-  0.183  -  0,111  =  14GOO  Ibs. 


18  RETAINING-WALLS  FOR  EARTH. 

From  (Vet), 

tan  d  =  o^lij  +  '577  =  ~  °-123  =  tan  <-  7°)' 

3°.  Algebraic  determination  of  the  value  ofB  tinder  Hie 
assumption  that  Q  =  \B. 

Substituting  the  proper  values  in  (11)  and  remembering 
that  ex  is  negative, 


B  -  -  4.7  ±  V163.44  +  (4.7)'  =  9.0  ft. 

The  foundation  can  be  designed  in  the  manner  outlined 
in  Ex.  1. 

Ex.  3.  Determine  the  dimensions  of  a  brick  wall  hav- 
ing a  vertical  back  to  retain  a  bank  of  sand  sloping  up- 
Avard  at  an  angle  of  20°.  0  =  30°,  H=  20',  B'  =  2', 
y  =  100. 

1°.  Algebraic  determination  of  E  and  d. 

Since  a  —  0, 

E=^A   .......     (2) 

400X.OO 


Xi 

The  value  of  A  is  readily  found  from  Diagram  I. 
d  =  e  =  20°,     since     a  =  0. 

2.  Algebraic  determination  of  the  value  of  B  under  the 
condition  ihat  Q  —  ^B. 


iATjl  1  9  7T7 

J5L  sin  *  +  B'  }  ==  ^  cos  S  +  B'\     (9) 


FORMULAS  FOR  EARTH  PRESSURE.  19 

From  Table  I,  W  =  125  Ibs.     Then 


or  B*  4-  6.65Z?  =  131.84. 


B  =  —  3.36  ±  V  131.84:  +  3.36*, 
and 

B  =  —  3.3G  +  11.96  =  7.60  ft. 

Ex.  4.    Determine  the  value  of  B  in  Ex.  3  under  the 
assumption  that  e  =  0  (horizontal  earth-surface). 


or  #  =  20000  (0.333)  =  6666,  say  6700  Ibs. 
Since  a  —  0,  and  e  =  0,  #  =  0, 


(10) 


B?  _j_9,#  =  111.2; 
B  =  -1  ±  Vlll.2  +  1, 
and  B  =  -1  -f-  10.59  =  9.6  ft. 

Ex.  5.    Determine  the  value  of  B  in  Ex.  3,  under  the 
assumption  that  e  —  0  =  30°. 

E  =  ^rcos  0  =  20000  (0.866)  =  17320  Ibs. 

Z 

From  (9), 


20  RETAINING- WALLS  FOR  EARTH. 

B'  +  15.865  =  244.05; 

B  =  -  7.93  +  A/244.05  +  T^jj?. 
and      B  =  —  7.93  +  17.52  =  9.6  ft. 

Ex.  6.  Determine  the  resultant  pressure  against  the 
back  of  a  wall  when  the  surface  of  the  earth  carries  a 
load  equivalent  to  5  feet  in  depth  of  sand. 

H  =  30  ft.,  a  =  10°,  0  =  30°,  e  =  0,  and  y  =  100 
Ibs. 


FIG.  6. 

Graphical  solution  of  the  problem. — In  Fig.  6,  let  B8 
represent  the  surface  of  the  earth,  and  BA  the  back  of 
the  wall. 

Make  ST  =  5,  and  draw  HT  and  BH.  Draw  AR  par- 
allel to  BS,  parallel  to  HT,  and  make  LR  equal  to  L  T; 
lay  off  the  angle  LRP  equal  to  30°;  with  Q  as  a  centre 


FORMULAS  FO&  EAHTH-pRmsutiE:.  21 

draw  an  arc  passing  through  L  tangent  to  PR,  and  then 
with  OR  as  a  radius  describe  the  circumference  of  the 
circle  RQM,  and  at  M  draw  MN  parallel  to  AH  \  at  A 
and  normal  to  AH  draw  A  C  equal  to  NL.  Then 


. 

The  direction  of  E  will  be  parallel  to  QM. 

To  determine  the  point  of  application  of  E,  find  the 
centre  of  gravity  E'  of  ABVC,  and  draw  E'D  parallel  to 
AC,  then  D  will  be  the  point  of  application  of  E. 

E'  can  be  found  as  follows:  Produce  A  C  and  BV,  make 
A  I  =  CK  =  B  V,  BG  —  VF—  AC,  and  join  ^  and  /  and 
G  and  K.  Then  .£",  the  intersection  of  Fl  and  $/r,  will 
be  the  centre  of  gravity  of  AB  VC.  BD  can  be  found 
from  the  formula 


c    10_ 
5 

See  (30)  of  Appendix. 

Ex.  7.  Determine  graphically  the  value  of  E  when  e  =  0 
and  a  =  0,  0.  y,  and  //  being  given. 

In  Fig.  7  let  /?.F  represent  the  surface  of  the  earth,  and 
AB  the  back  of  the  wall.  Draw  AL  parallel  to  BF  and 
make  IL  =  IF',  lay  off  the  angle  GLH  =  0,  and  at  any 
point  K  in  LH  draw  JOT  perpendicular  to  HL,  and  lay 
off  JfO  =  MK\  draw  J//  parallel  to  01.  Then  will  the  arc 
IN,  described  with  J  as  a  centre  and  IJ  as  a  radius,  puss 
through  /  and  be  tangent  to  GL\  with  J  as  a  centre  and 
JL  as  radius  describe  the  circumference  LH  ;  at  ^4  lay  off 
.4(7  =  ///and  normal  to  ^47?.  Then 


V2  RETAINING-  WALLS  FOR 

E  \&  parallel  to  .Z?.Fand  applied  at  D,  AD  being  equal  to 


FIG.  7. 

Ex.  8.  Determine  the  earth-thrust  on  the  profile  shown 
in  Fig.  8,  H,  y,  <p,  and  e  being  given. 

Graphical  solution  of  lite  problem. — Let  BCDEA  repre- 
sent the  given  profile,  and  let  the  surface  of  the  earth 
be  horizontal.  Prolong  BC  until  it  intersects  SA  in  S; 
draw  SR  normal  to  #<7$and  equal  to  the  intensity  of  the 
earth-pressure  at  8',  connect  B  and  R.  Then  from  the 
middle  point  of  B C  draw  GF  parallel  to  SR\  the  distance* 
GF  multiplied  by  y  will  be  the  average  intensity  of  the 
earth-pressure  on  BC.  In  a  similar  manner  the  average 
intensities  on  CD,  DE,  and  EA  can  be  found,  and  hence 
the  total  pressures  on  each  determined.  The  points  of  ap- 
plication of  these  resultant  pressures,  El9  E9,  Ez,  and  E^9 


FORMULAS  FOR  EARTH-PRESSURE. 


Can  be  found   by  the  method  used  in  Ex.  6  for  finding 
the  centre  of  gravity  of  a  trapezoid.     The   directions  of 
B 


\  AX^^SSS^^^ 

k 


FIG.  8. 

E^ ,  E^ ,  ES  ,  and  EI  are  found  from  the  construction  on  the 
right. 

Ex.  9.  Determine  the  thrust  of  the  earth  against  a  ver- 
tical wall  when  e  is  negative. 

For  the  explanation  of  this  construction,  see  Part  II, 
page  47,  Fig.  Sa. 

Ex.  10.  From  the  following  data  determine  E,  d,  and 

Q- 

e  =  0,  0  =  38°,  a  =  10°  23';  y  =  90  Ibs.,  W=  170  Ibs.j 

H=  15  ft.,     B  =  §  ft.,     B'  =  2  ft. 
Ans.  E  =  3037  Ibs.,  6  =  27°  13',  Q  =  2.2  ft. 

Ex.  11.  Determine  the  dimensions  of  a  trapezoidal  wnlj 
built  of  dry,  rough  granite,  having  a  vertical  back  and 
being  20  feet  high,  to  safely  retain  the  side  of  a  sand  cut, 


Fofi  E 


the  surface  of  the  sand  being  level  with  the  top  of  the  wall. 
W—l^  Ibs.,  y=WO  Ibs.,  0  =  33°  40',  H  =  20  ft., 
B'  =  2  ft. 

Ans.  E  =  5734  Ibs.,  3  =  0,  B  —  8  ft.,  and   Q  =  2.8  ft., 
about. 

Ex.  12.   The  same  as  Ex.  11,  with   a  =  8°  instead    of 
or  =  0. 

Ans.  E  —  6330  Ibs.,  B  =  8  ft.,  and  §  =  2.7  ft. 


FIG.  8a. 

Ex.  13.  What  must  be  the  dimensions  of  a  rubble  wall 
of  large  blocks  of  limestone,  laid  dry,  to  retain  a  sand 
filling  which  supports  two  lines  of  standard-gauge  railroad 
track  ?  (Assume  the  depth  of  sand  to  produce  a  pressure 
on  the  earth  equal  to  that  produced  by  the  railroad  and 
trains  as  4  feet.) 


FORMULAS  FOR  EARTH- PRESSURE.  25 

H  =  15  ft.,  a  =  8°,  0  =  33°  40',  y  =  100  Ibs.,  JF  =  170 
Ibs.,  J5'  =  3.5  ft. 

^s.   ^=5760  Ibs.,  $  =  18°  7',  5  =  8  ft,  Q  =  2.7ft. 

Ex.  14.  Determine    ^,  tf,  £,   and    ft  when    W  =  170 
Ibs.,  7  =  100  Ibs.,  a  =  8°,  e  =  0  =  33°   40',  H  =  20  ft., 
£'  =  2  ft. 
A  us.  E  =  21760  Ibs.,  d  =  32°  25',  5  =  9  ft.,  Q  =  3  f  t. 

*  Ex.   15.  A  wall  9  ft.  high  faces  the  steepest  declivity 
of  earth  at  a  slope  of  20°  to  the  horizon;  weight  of  earth 
130  Ibs.  per  cubic  foot,  angle  of  repose  30°.     Determine 
E  when  a  =  0. 

Ans.  E  =  2 187  Ibs. 

*  Ex.  16.   e  =  33°  42',   0  =  36°,   H  =  3  ft.,  y  =  120 
Ibs.,  a  =  0.     Determine  E. 

Ans.  E  =  278  Ibs. 

*  Ex.   17.   0  =  25°,   e  =  0,   a  —  0,  H  =  4  ft.,  y  =  120 
Ibs.,  E=  ? 

Ans.  E=  390  Ibs. 

*  Ex.  18.    0  =  38°,    e  -  0,    a  =  0,  H  =  3  ft.,    y  =  94 
Ibs.,  E  =  ? 

Ans.  E  =  100.5  Ibs. 

*  Ex.  19.  A  ditch  6  feet  deep  is  cut  with  vertical  faces 
in  clay.     These  are  shored  up  with  boards,  a  strut  being 
put  across   from  board   to  board  2  feet  from  bottom,  at 
intervals   of   5  feet  apart.     The  coefficient  of  friction   of 
the  moist  clay  is  0.287,  and  its  weight  120  Ibs.  per  cubic 
foot.     Find  the  thrust  on  a  strut,  also  find  the  greatest 
thrust  which  might  be  put  upon  the  struts  before  the  ad- 
joining earth  would  heave  up. 

Ans.  E  —  1230  Ibs. 
Thrust  per  strut  =  6128  Ibs. 
Greatest  thrust  =  19029  Ibs. 

*  Alexander's  Applied  Mechanics. 


26  RETAINING- WALLS  FOR  EARTH. 

*  Ex.  20.  A  wall  10  ft.  high,  2  ft.  thick,  and  weighing 
144  Ibs.  per   cubic  ft.,  is  founded  in  earth  weighing  112 
Ibs.  per  cubic  ft.,  and  whose  angle  of  repose  is  32°.     Find 
the  least  depth  of  the  foundation. 

Aits,  x'  =  1.21   ft.     10  -  1.21  =  8.79  ft.  =  amount   of 
wall  above  the  ground. 

*  Ex.  21.  An  iron  column   is  to  bear   a  weight  of   20 
tons   (2240  Ibs.  =  one  ton);  the  foundation  is  a  stone  3 
ft.  square  on  bed,  sunk  in  earth  weighing  120  Ibs.  per  cu. 
ft;  angle  of  repose  27°.     Find  the  least  depth  to  which  it 
must  be  sunk  for  equilibrium. 

Ans.  x'  =  6  ft. 

*  Ex.  22.  A  brick  wall,  allowing  for  openings,  weighs 
42000  Ibs.  per  rood  of  36  sq.  ft.  (on  an  average  one  brick 
and  a  half),  and   stands   45   ft.   above  the   ground;   the 
foundation  is  to  widen  to  four  bricks  at  the  bottom.     Find 
depth  of  foundation  in  clay  weighing  130  Ibs.  per  cu.  ft. 
(angle  of  repose  27°),  neglecting  weight  of  unknown  foun- 
dation. 

Ans.  x'  =  1.7  ft. 

*  Alexander's  Applied  Mechanics. 


PART  II. 


THE  THEORY   OF   EARTH-PRESSURE   AND  THE 
STABILITY  OF  RETAINING-WALLS. 

Preliminary  Principles. — Before  demonstrating  the  gen- 
eral formula  for  the  thrust  of  earth  against  a  wall,  it  will 
be  necessary  to  establish  the  relations-  between  the  stresses 
in  an  unconfined  and  homogeneous  granular  mass. 

*  In  Fig.  1  let  ABC\)Q  any  small  prism  within  a  granu- 

F        Q  P 


H 


FIG.  1. 

lar  mass  which  is  in  equilibrium  un  er  the  action  of  the 
three  stresses  P,  Q,  and  R,  having  the  intensities  p,  q,  and 
r  respectively. 

*  In  all  the  demonstrations  which  follow,  the  dimension  perpen- 
dicular to  the  page  will  be  considered  as  unity. 

27 


28  HETA1N1NG-WALLS  FOR  EARTH. 

Let  6  represent  the  angle  of  inclination  of  the  plane  CB 
with  A  By  and  the  angle  at  A  be  a  right  angle. 

The  planes  AB  and  AC  are  called  planes  of  principal 
stress,  and  P  and  Q  are  called  principal  stresses. 

CASE  I.  If  the  principal  stresses  are  of  the  name  kind 
and  their  intensities  the  same)  then  will  the  resultant  stress 
on  any  third  plane  he  normal  to  that  plane  and  its  inten- 
sity he  equal  to  that  of  either  principal  stress. 

In  Fig.  1,  for  convenience,  let  AB  =  1,  then  AC  —  tan  0, 

and  CB  = 7,.     Hence 

cos  6 

7* 

P  =  p,  Q  =  q  tan  0  =  p  tan  8,  since  p  =  q,  and  72  = „ . 

cos  u 

Since  P,  Q,  and  72  are  in  equilibrium,  they  will  form  a 
closed  triangle,  as  shown  on  the  right  in  Fig.  1.  Hence 

72s  =  P8  +  C2, 
or 

=  f  +p*  tan2  0  =^(1  +  tan'  0); 


.  • .  r  =  p  =  q. 
Also,  RcoaFDE=P, 


or  —^-7,  cos  7^7)7?  =  p ;    but  r  =  p. 

cos  0 


Hence  cos  0  =  cos  .FTXE7  =  cos  777)  #; 

=  0    and     72  is  normal  to  CB. 


THEORY  OF  EARTH-PRESSURE.  29 

CASE  II.  If  the  principal  stresses  are  not  of  the  same 
kind  Int  their  intensities  the  same,  then  will  the  resultant 
make  the  angle  6  with  the  direction  of  t lie  principal  stress, 
but  on  the  opposite  side  from  that  on  which  the  resultant  in 
Case  I  lies,  and  its  intensity  he  equal  to  that  of  either  prin- 
cipal stress. 

The  demonstration  of  Case  I  proves  this  principle  if 
Fig.  1  is  replaced  by  Fig.  2. 


FIG.  2. 

CASE  III.  Given  the  principal  stresses  of  the  same  kind 
l)iit  having  unequal  intensities,  to  determine  the  intensity 
and  direction  of  the  resultant  stress  on  any  third  plane. 

Let  P  and  Q  be  compressive  and  the  intensity^  >  the 
intensity  q. 

The  following  identities  can  be  written: 


P  =  UP  +  9)  +  l(p  -  q), 


and 


30 


RETAINING-WALLS  FOR  EARTH. 


or  the  resultant  intensity  on  the  plane  CB  may  be  con- 
sidered as  being  the  resultant  of  two  intensities,  one  being 
the  intensity  of  the  resultant  stress  caused  by  two  like  prin- 
cipal stresses  having  the  same  intensity  |(;j  -[-  q),  and  the 
other  the  intensity  of  the  resultant  stress  caused  by  two 
unlike  principal  stresses  having  the  same  intensity  %(p  —  ({)• 


FIG.  3. 

The  intensity  of  the  resultant  stress  caused  by  the  first 
two  principal  stresses  will  be,  by  Case  I,  \(p  +  </),  and  the 
direction  of  the  resultant  will  be  normal  to  the  plane  CB. 
By  Case  II  the  resultant  of  the  second  pair  of  principal 
stresses  will  make  the  angle  6  with  the  direction  of  P,  and 
its  intensity  will  be  ^(p  —  q);  then  the  resultant  intensity 
can  be  found  as  follows: 

In  Fig.  3  draw  MD  normal  to  BC,  and  make  LD  = 
^(p-\-q)',  with  L  as  a  centre  and  LD  as  radius,  describe 
an  arc  cutting  FD  at  F.  Then  the  angle  LFD  =  LDF  =  6. 
Lay  off  LG  =  $(p  —  q),  and  draw  GD,  which  is  the  result- 


THEORY  OF  EARTH-PRESSURE.  31 

ant  intensity,  and  the  intensity  of  the  resultant  stress  on 
CD  caused  by  the  two  principal  stresses  P  and  Q.  GD 
also  represents  the  direction  of  the  resultant  stress  R. 

Since  the  intensities  of  the  principal  stresses  remain  con- 
stant, %(p  -f-  q)  and  l(p  —  q)  will  remain  the  same  for 
any  inclination  of  the  plane  GB\  hence  the  intensity  r  of 
the  resultant  depends  upon  the  angle  6  when  p  and  q  are 
given. 

From  Fig.  3, 

GLcos2&=LM    and     GLsin2V=GM, 
DM  =  DL  +  LM  =  t(p  +  q)  +  \(p  -  q)  cos  20, 


or 


r  =  Vp*  cos2  0  +  </2  sin*  0,  .    .     .    .    (a) 


which  is  the  general  expression  for  the  intensity  of  the 
resultant  stress  of  a  pair  of  principal  stresses. 

As  the  angle  6  changes,  the  angle  ft  will  also  change, 
and  it  will  have  its  maximum  value  when  the  angle 
LGD  —  90°.  This  is  easily  proven  as  follows: 

With  L  as  centre  and  GL  as  radius  describe  an  arc; 
then  ft  will  have  its  maximum  value  when  the  line  DG  is 
tangent  to  the  arc;  but  when  DG  is  tangent  to  the  arc  the 
angle  LGD  is  a  right  angle,  since  LG  is  the  radius  of  the 
arc. 


(b) 


from  which  the  following  can  be  easily  obtained  : 

jo  _  1  —  sin  max  /? 
~q  ~  I  +  sin  max"/?' 


32 


RETAINING- WALLS  FOR  EARTH. 


which,  expresses  the  limiting  ratio  of  the  intensities  of  the 
principal  stresses  consistent  with  equilibrium,  p  being 
greater  than  q. 

CASE  IV.  Given  the  intensity  and  direction  of  the  re- 
sultant stress  on  any  plane,  and  the  value  of  max  ft,  to 
determine  the  intensities  and  directions  of  the  principal 
stresses. 


Fm.  4. 


Let  AD  represent  the  given  plane  and  GD  the  direction 
and  intensity  of  the  resultant  stress  at  the  point  D. 

Draw  DL  normal  to  AD,  and  draw  DI,  making  the  angle 
max  /3  with  LD.  At  any  point  J  in  DL  describe  an  arc 
tangent  to  DI,  cutting  GD  in  K  and  draw  GL  parallel 
to  KJ\  with  L  as  a  centre  and  L  G  as  radius  describe 


THEORY  OF  EARTH-PRESSURE.  33 

a  circumference.     This  circumference  will  pass  through  G 

GL 

and  be  tangent  to  DI\  hence  -,-ry  =  sin  max  /?. 

Since  sin  max  ft  =  -  -  -  ,  and   GL  and  LD  are  com- 

P  +  V 
ponents  of  r, 

GL  =  l(p-q)     and     DL  =  ±(p  +  q); 
then  ND 
and      MD  =  LD-  LM  = 


which  completely  determines  the  intensities  of  the  principal 
stresses. 

According  to  Case  III,  the  direction  of  the  greater  prin- 
cipal stress  bisects  the  angle  between  the  prolongation  of 
LM  and  the  line  GL;  hence  RL  represents  the  direction 
of  the  greater  principal  stress,  and  that  of  the  other  is  at 
right  angles  to  RL. 

The  above  intensities  and  directions  being  determined, 
the  intensity  of  the  resultant  stress  on  any  other  plane 
passing  through  D  is  easily  determined  as  follows: 

Let  DY  represent  any  plane  passing  through  D,  draw 
DL'  normal  to  MY  and  equal  to  %(p  -f  q).  Draw  R'D 
parallel  to  RL,  and  with  L'  as  a  centre  and  L'D  as  radius 
describe  an  arc  cutting  R'D  at  0,  and  make  L'G'=  l-(p—q)'9 
then  G'D  =  r'  =  the  intensity  of  the  resultant  stress  on 
DY. 

It  is  clear  that  if  the  value  of  max  /3  can  be  obtained 
for  a  mass  of  earth  that  the  construction  of  Fig.  3  can  be 
employed  in  determining  the  intensity  of  the  earth-pressure 
at  any  point  in  dn^  plane  within  the  mass. 


34 


RETAINING- WALLS  FOR  EARTH. 


It  has  been  established  by  experiment  that  if  a  body  be 
placed  upon  a  plane,  that  (as  the  plane  is  made  to  incline 
to  the  horizontal)  at  some  angle  of  inclination  the  body 
will  commence  to  slide  down  the  plane,  and  that  this  angle 
depends  largely  upon  the  character  of  the  surfaces  in  con- 
tact. 


B 


FIG.  5. 

In  Fig.  5  let  AS  represent  a  plane  inclined  at  the  angle 
0  with  the  horizontal,  and  C  any  mass  just  on  the  point  of 
sliding  down  the  plane.  Let  EC  represent  the  weight  of 
the  mass  C,  and  ED  and  DC  the  components  respectively 
parallel  and  normal  to  the  plane  AB.  Then  DE  is  the 
force  required  to  just  keep  the  mass  C  from  sliding  down 
the  plane,  assuming  the  plane  to  be  perfectly  smooth,  or  if 
the  plane  is  rough  this  force  represents  the  effect  of  fric- 
tion. 

DE 


or  when  the  mass  C  is  about  to  slide,  the  resultant  pres- 
sure EC  on  A£  makes  the  angle  <p  with  the  normal  to  the 


THEORY  OF  EARTH- PRESSURE.  35 

plane,  the  angle  0  being  the  inclination  of  the  plane  AB, 
and  is  called  the  angle  of  friction. 

In  the  case  of  earth,  considered  as  a  dry  granular  mass, 
the  inclination  of  the  steepest  plane  upon  which  earth  will 
not  slide  is  called  the  angle  of  repose,  and  the  plane  the 
surface  of  repose. 

From  the  above,  then,  it  follows  that  in  a  mass  of  earth 
the  resultant  pressure  on  any  plane  cannot  make  an  angle 
with  the  normal  to  that  plane  which  is  greater  than  the 
angle  of  repose  0 ;  therefore  the  construction  of  Case  IV 
applies  to  earth  when  max  ft  is  replaced  by  0.  The  values 
of  0  for  earth  under  various  conditions  are  given  in 
Table  II. 

The  preceding  principles  will  now  be  applied  in  deter- 
mining the  thrust  of  earth  against  a  retaining- wall. 


EARTH-PRESSURE. 

In  order  that  the  formulas  may  not  become  too  complex 
for  practical  use,  it  will  be  assumed  that  the  earth  is  a 
homogeneous  granular  mass  without  cohesion.  The  surface 
of  the  earth  will  be  considered  to  be  a  plane,  and  the  length 
of  the  mass  measured  normally  to  the  page  as  unity. 

*  Given  the  intensity  and  direction  of  the  resultant  stress 
at  any  point  in  any  plane  parallel  to  the  surface  of  the 
earth,  the  inclination  of  the  surface  of  the  earth  with  the 
horizontal,  and  the  angle  of  repose,  to  determine  the  in- 
tensity and  direction  of  the  resultant  stress  on  a  vertical 
plane  passing  through  the  same  point. 

*For  comparison,  see  the  "  Technic,"  1888;  a  construction  by 
Prof.  Greene. 

The  construction  follows  (see  Fig.  4,  above)  directly  from  Rau- 
kiue's  Ellipse  of  Stress. 


36 


RETAIN1NG-WALLS  FOR  EARTH. 


In  Fig.  6  let  B  Q  represent  the  surface  of  the  earth,  and 
D  any  point  in  the  plane  AD  parallel  to  BQ  ;  draw  DQ 
normal  to  AD,  and  make  the  vertical  GD  equal  to  QD; 
then  OD-y  is  the  intensity  of  the  resultant  pressure  at  D. 
Draw  DM,  making  the  angle  0  with  LD,  and  with  L  as 
centre  describe  an  arc  tangent  to  DJf  and  passing  through 
G\  then  by  Case  IV  LG-y  =  \(p  -  q),  LD.y  =  $(p  +  q), 


FIG.  6. 


and  RL  bisecting  the  angle  QLG  is  the  direction  of  the 
greater  principal  stress.  To  determine  the  intensity  and 
direction  of  the  resultant  stress  at  D  on  a  vertical  plane, 
proceed  according  to  Case  IV.  Draw  R'D  parallel  to  RL 
and  DL'  =  DL  normal  to  DG.  With  L'  as  a  centre  and 
L'D  as  radius  describe  an  arc  cutting  R'D  at  R"3  and  make 


THEORY  OF  EARTH-PRESSURE.  87 

L'G'  =  LG\  then  DG'  represents  the  direction  of  the 
resultant  stress,  and  DG'-y  the  intensity  of  the  resultant. 

In  Fig.  6  the  angle  R'DL'  =  DR"L'  =  90°  -  GO  +  0'. 
.-.  O'L'D  =  2oo-  2V'.  But  20'  =  GO  +  e;  hence  G'L'D 
=  GO  —  e. 

Draw  LY  =  LG',  then  the  angle  D  LY  =  GO  —  e.  /.Since 
LD  =  DL'  and  LY=  LG  =  L'G',  the  triangle  O'L'D 
equals  the  triangle  LYD  and  the  angle  G'DL'  =  e;  or  the 
direction  of  the  resultant  earth-pressure  against  a  vertical 
plane  is  parallel  to  the  surface  of  the  earth. 

From  Fig.  6, 


—  q  cos  <*>  —       >  y, 

~  $)  siQ  °°  —  LX  •  y, 
+  %)  cos  e  —  DX*  y. 

Now  DY=DG'  =  DG-2GX, 

or 

DG'  -  y  =  DG  -  y  —  (p  —  q)  cos  G? 

=  UP  +  0)  cos  e  -  ±(p  -  q)  cos  GO, 
\(P  +  4)  •  sil1  <*>     ::     i(/>  -  y)  :  sin  e, 
and 

p  -\-  q    • 
sin  09  =  ^-  !  —  *•  sin  e, 

P-flf 

or 


At  •  .       A 

cos  GL>  =  i/  1  —  K—  i-i     sin2  e  =       ^ 


/  x 

\P  -  fiv  (^  -  fl') 

and  since  J(p  +  g')  sin  0  =  J(^  —  g'), 


cos  oo  =  - — -  f/cos2  e  —  cosa  0. 
gin  0 


38  RETAINING-WALLS  FOR  EARTH. 

Substituting  this   value  for  cos  GO  in  the  equation  for 
DG' .  y,  it  becomes 

1       

DG'  -y  =  %(p  +  q)  cos  e  —  l(p— q)  —. — -.  1/cos2  e  —  cos2  0, 

u  sin  0 

1          p-\-q 
or  since  -  * 


-r—  -      —  — . 
sin  0      p  —  q 


DG' .  Y  —  l(p  +  q)  !cos  6  —  ^cos3  e  —  cos2  0J. 
In  a  similar  manner, 


e  -  cos2 
and 


DG'    _  cos  e  —  4/cos2  e  —  cos2  0 
DG        cos  e  -j-  4/cos2  e  —  cos2  0 
hence 


T^nr  r>  ^COS  e  —  4/COS2  €  —  COS2  0 

JJ(JT   = 


cos  e  -{-  I/cos2  e  —  cos2  0 

Let  x  =  the  vertical  distance  between  the  two  planes 
and  ADt  then 

cos  e  —  Vcos2  e  —  cos2  0 


.-.DG'-y  =  (x)  y  cose 


cos  e       v  cos    e  —  cos 


which  is  the  expression  for  the  intensity  of  the  resultant 
earth-pressure  on  a  vertical  plane  at  any  depth  x  below  the 
surface. 
Let 


cos  e  —  Vcos2  e  —  cos    0 

*  A  =  cos  e  -  — —  =.     .     .     (d) 

cos  e  -f-  T  cos2  e  —  cos2  0 

*  See  Rankine's  Applied  Mechanics ;    Alexander's  Applied  Me- 
chanics ;  Theories  of  Winkler  and  Mohr. 


THEORY  OF  EARTH-PRESSURE.  39 

The  average  intensity  of  the  resultant  earth-pressure  on 
a  vertical  plane  of  the  length  x  will  be 


and  hence  the  total  pressure  will  be 


Since  the  intensities  of  the  pressures  are  uniformly  varying 
from  the  surface,  and  increasing  as  x  increases,  the  appli- 
cation of  the  resultant  thrust  will  be  at  a  depth  of  \x  be- 
low the  surface. 

Considering  the  earth  as  an  unconfined  mass,  the  above 
formula  is  perfectly  general  and  can  be  applied  under  all 
conditions,  including  the  case  when  e  is  negative. 

The  resultant  stress  on  any  plane  as  AB,  Fig.  6,  can  be 
found  by  applying  the  principles  of  Case  IV.  Draw  PA  • 
parallel  to  RL,  make  AN=  ZZ)and  NO  =  LG\  then  AO 
represents  the  direction  of  the  resultant  pressure  on  AB. 
Make  AC  =  AO',  then  the  area  of  the  triangle  ABC  mul- 
tiplied by  y  is  the  total  pressure  on  the  plane  AB,  and  this 
pressure  is  applied  at  \AE  below  B. 

In  unconfined  earth  this  construction  is  perfectly  gen- 
eral and  applies  to  any  plane.  It  also  applies  equally  well 
to  curved  profiles.  An  example  illustrating  the  applica- 
tion of  the  method  will  be  given  in  the  applications.  See 
pages  22  and  23. 

The  following  graphical  construction,  Fig.  7,  is  more  con- 
venient than  that  of  Fig.  6. 

As  before,  let  BE  represent  the  surface  of  the  earth,  and 


40 


RETAINING-WALLS  FOR  EARTH. 


AD  a,  plane  parallel  to  the  surface.  At  any  point  D  in 
this  plane,  draw  DE  vertical  and  make  DF  —  DE  \  draw 
FG  horizontal  and  make  the  angle  HFD  =  0. 

With  L  as  a  centre,  describe  an  arc  passing  through  G 
and  tangent  to  MF',  then  with  L  as  a  centre  and  LF  as 


B| 


FIG.  7. 


radius,  describe  the  circumference  FON,  cutting  AD  at  N; 
through  N  draw  NO  parallel  to  AB,  then  draw  AC  nor- 
mal to  AB  and  equal  to  OG.  The  area  of  the  triangle 
AB C  multiplied  by  y  will  be  the  total  earth-pressure  on 
AB.  To  determine  the  direction  of  the  thrust  prolong  OG 
to  ft  then  QN  is  the  direction  of  the  thrust. 

That  this  construction  is  equivalent  to  that  of  Fig.  6  is 


THEORY  OF  EARTH-PRESSURE.  41 

proved  as  follows.     The  triangle  GLF  of  Fig.  7  equals  the 
triangle  OLD  of  Fig.  6. 


.   GL'Y  —  \(p  —  q]      and     LF-y  =  LO-y  =  {(p  +  q). 
In  Fig.  6,  the  angle  NAP  =  NPA  =  90°—  $(GO—  e)  —  a. 

.  •  .   ONA  =  GO  —  e  +  2a. 
In  Fig.  7,  the  angle  OLN=2e-2a.     But  GLN=  c»+e. 


and  GO  of  Fig.  7  equals  A  0  of  Fig.  6. 

In  Fig.  7,  the  angle  QNO  =  90°  -  /?'. 

In  Fig.  6,  the  angle  GAB  =  90°  -  ft'. 

Therefore  the  direction  of  the  thrust  is  the  same  in  both 
constructions. 

The  two  constructions  given  above  are  all  that  is  re- 
quired to  determine  the  thrust  of  earth  upon  any  plane 
within  the  mass  of  earth,  as  one  can  be  used  as  a  check 
upon  the  other;  but  as  a  formula  is  often  very  convenient, 
a  general  formula  will  now  be  deduced  which  will  enable 
one  to  determine  the  values  of  E  and  d  for  any  plane  with- 
in a  mass  of  earth. 

GENERAL  FORMULA  FOR  THE  THRUST  OF  EARTH. 

In  Fig.  8,  let  BQ  represent  the  surface  of  the  earth  and 
AB  any  plane  upon  which  the  earth-pressure  is  desired. 
Draw  AD  parallel  to  BQ  and  let  the  vertical  distance 


42 


RETAINING-WALLS  FOR  EARTH. 


From  (e)  the  earth-pressure  upon  FA  is  parallel  to  the 
surface  and  equal  to 


FIG.  8. 


But  AF=  x  =  //(I  +  tan  a  tan  e)  = 


cos  a  cos  e 


2     cos 


COS8  (6  -  Of) 

2  '     '     ' 


Now   the   thrust  P   combined  with  the  weight  of  the 
prism  ABFmn&i  produce  the  resultant  pressure  upon  AB. 


THEORY  OF  EARTH-PRESSURE.  43 

Then  from  Fig.  8, 

V  —  —~  tan  a  (1  -f  tan  a  tan  e) 

6 

H*y  sin  a  cos  (e  —  a) 


E  =  V(  V+P  sin  e)a+(P  cos  e)a  =  I/  F2+Pa+  2  FP  sin  e. 
Substituting  (/)  and  (g)  in  this  it  becomes 


jBV  cos  (e  -  a) 

-tv  =  -  —    -  ^  --  X 

2    cos*  a  cos  e 


/  ~A  A* 

4/sin2  a  4-  2  sin  a  sin  e  cos  (e  —  a)  --  1-  cos2  (e  —  a)  —  —  , 

cos  e   '  x  cos-  e 

which  becomes,  by  replacing  A  by  its  value  from  (d  ), 


2     cos2  a  cos  e 


-}-  sin2  a 


cos  e—  i/cos8  e—  cos2 
4-  2  sin  nr  sin  e  cos  (e—  a) 


cos  e-j-  y  cos2  e— cos2  <£>  /j\ 

,  cos  e  —  I/cos2  e— cos2  (p  )  2 
cos2  (e  —  a)  ]  — ; — 

(  cos  e  -f-  \  cos2  e— cos-  0  ) 

which  is  the  general  equation  for  the  thrust  of  earth  upon 
any  plane  within  the  mass. 

To  determine  the  direction  of  the  thrust  of  the  earth, 
let  8  be  the  angle  which  the  direction  of  the  thrust  makes 
with  the  horizontal;  then,  from  Fig.  8, 

y 

i&n  6  =  -=-     -  -f  tan  e. 
P  cos  e   ' 


44  RETAINING-WALLS  FOR  EARTH. 

Substituting  the   values   of    V  and  P  given   above,  this 
becomes 


.       sin  a  cos  e  -f  sm  e  cos  (e  —  a]  A 

tan  d  =  - '- — .    .     (la) 

cos  e  cos  (e  —  a)  A 


where 


cos  e  —  Vcos2  e  —  cosa  0 

A  =  cos  e  —  =.     .    .      (r/) 

cos  e  -j-  fcosa  e  —  cos2  0 

Equations  (1)  and  (la)  are  readily  reduced  to  more  sim- 
ple forms  for  special  cases.  These  forms  will  be  found  in 
Part  I. 

The  Plane  of  Rupture. — Although  it  is  not  necessary  to 
know  the  position  of  the  plane  of  rupture  in  order  to  deter- 
mine the  thrust  of  the  earth,  yet  it  may  be  of  interest  to 
know  its  position,  which  can  be  easily  determined  as  fol- 
lows : 

The  plane  of  rupture  will  be  back  of  the  wall  and  pass 
through  the  heel  of  the  wall.  The  resultant  earth-pressure 
will  make  the  angle  0  with  the  normal  to  this  plane.  Now 
the  tangent  of  the  angle  which  the  direction  of  the  result- 
ant earth -pressure  on  any  plane  makes  with  the  horizontal 
is  determined  from  the  formula 

sin  a 

tan  o  = -. —    — r— r  ~\-  tan  e. 

cos  (e  —  oi)A 

If  GO  represents  the  angle  which  the  plane  of  rupture  makes 
with  the  vertical  passing  through  the  heel  of  the  wall, 
a  =  GO  and  d  =  0  -J-  GO. 

tan  (0  +  GO)  =  -. r-7  +  tan  e, 

cos  (e  -  GO)  A 

from  which  the  value  of  GO  can  be  determined  for  any  case. 


THEORY  OF  EARTH-PRESSURE.  46 

For  the  case  where  e  =  0,  e  being  positive  with  respect 
to  the  wall  and  negative  with  respect  to  the  plane  of  rupture, 
the  above  equation  becomes 

sin  a> 

tan  (0  4-  G?)  =  ---  T—  --  —7  —  tan  0, 

cos  (0  +  a?)  cos  0 

which  is  satisfied  when  GO  =  90°  —  0. 
For  the  case  where  e  —  0, 


sn 
tan  (0  +  GO)  =  - 


cos  GO  tan2 


which  is  satisfied  when  G?  =  45°  —  —  . 


Reliability  of  the  Preceding  Theory.  —  The  preceding 
theory  is  based  upon  the  assumptions  that  the  earth  is  a 
homogeneous  mass  and  without  cohesion,  and  the  formulas 
are  deduced  under  the  assumption  that  the  surface  of  the 
earth  is  a  plane. 

All  writers  on  the  subject  have  considered  the  earth  as  a 
homogeneous  mass  and,  with  a  few  exceptions,  without 
cohesion. 

Old  and  recent  experiments  indicate  that  cohesion  has 
very  little  effect  upon  the  pressure  of  the  earth,  which  ex- 
plains why  it  has  not  been  considered  by  most  writers. 

The  assumption  of  a  plane  earth-surface  is  necessary 
whenever  practical  formulas  and  direct  graphical  construc- 
tions for  obtaining  the  thrust  of  the  earth  are  obtained. 
General  formulas  can  be  deduced  for  any  character  of  sur- 
face, but  they  are  too  complex  for  practical  use.  Tliose 
graphical  constructions  which  do  not  require  a  plane  earth- 


46       RETAINING-  WALLS  FO&  EARTH. 

surface  are  not  direct  in  tlieir  solution  of  the  problem,  but 
require  a  series  of  trials  to  obtain  the  maximum  thrust. 

If  the  earth-surface  is  not  a  plane,  one  can  be  assumed 
which  .will  give  the  thrust  of  the  earth  sufficiently  exact 
for  all  practical  purposes. 

For  uncon fined  earth  no  exceptions  can  be  taken  to  the 
preceding  theory,  the  assumptions  upon  which  it  is  based 
being  accepted,  and  for  confined  earth  the  theory  must  be 
true  when  the  direction  of  the  principal  stress  passing 
through  the  heel  of  the  wall  lies  entirely  within  the  earth. 

For  all  cases  in  which  a  and  e  are  positive  the  theories 
of  Rankine,  Winkler,  Weyraucli,  and  Mohr  agree  and  give 
identical  results  with  the  preceding  theory,  as  they  should, 
being  founded  upon  the  same  assumptions. 

When  a  is  negative  Weyraiich  does  not  consider  his 
theory  reliable,  and  his  equations  lead  to  indeterminate  re- 
sults. 

WinTcler  and  Mohr  consider  their  theories  reliable  when- 
ever the  direction  of  the  principal  stress  passing  through 
the  heel  of  the  wall  lies  entirely  within  the  earth. 

Rankings  method  of  considering  the  case  where  a  is 
negative  is  equivalent  to  assuming  that  the  introduction  of 
a  wall  does  not  affect  the  stresses  within  the  mass. 

It  may  be  concluded  that  the  preceding  theory  is  per- 
fectly exact  when  a  and  e  are  positive;  and  when  a  or  e  is 
negative  that  the  stresses  obtained  will  be  the  maximum 
which  under  any  circumstances  can  exist. 

For  the  case  where  e  is  negative  the  stress  obtained 
(which  represents  the  maximum  thrust  the  wall  can  have 
against  the  earth  and  have  equilibrium)  will  be  considerably 
larger  than  the  actual  stress  (when  a  wall  is  introduced), 
depending  upon  the  magnitude  of  e.  For  small  values  of  e 
the  results  will  be  practically  correct.  For  large  values  of  e 


THEORY  OF  EARTH-PRESSURE. 


47 


the  following  method  can  be  employed  in  determining  the 
thrust  of  the  earth.  The  method  depends  upon  the  assump- 
tion that  the  pressure  of  the  earth  is  normal  to  the  back  of 
the  wall.  This  may  or  may  not  be  the  case,  but  it  appears 
to  be  the  most  consistent  assumption  to  make  for  this  rare 
and  not  important  case. 


Fio.  8a. 

*  In  Fig.  80,  let  AB  be  the  back  of  the  wall  and  Bfi\\Q 
surface   of   the   earth.      Make   Ba  =  ab  =  be  =  cd  =  etc. 
Some  prism  BAa  or  BAb  or  BAc,  etc.,  will  produce  the 
maximum  thrust  on  the  wall;  and  when  this   maximum 
thrust  is  produced,  the  resultant  pressure  on  the  plane  Aa 

*  See  Van  Nostrand's  Magazine,  xvn,  1877,  p.  5.     "New  Con- 
structions in  Graphical  Statics,"  by  H.  T.  Eddy,  C.E.,  Ph.D. 


48  HETAININO-WALLS  FOR  EAHT8. 

or  A  b  or  A c,  etc.,  will  make  the  angle  0  with  the  normal 
to  the  plane. 

On  the  vertical  line  Ad'  la,yoB.Aa'=a'b'  =  b'c',  etc.,  and 
draw  Aa"  making  the  angle  0  with  the  normal  to  Aa,  Ab" 
making  the  angle  0  with  the  normal  to  Ab,  etc.;  then  draw 
a' a",  b'b",  etc.,  perpendicular  to  AB,  and  draw  a  curve 
through  Aa",  b",  c" ,  etc.  Then  there  will  be  a  maximum 
distance  parallel  to  a'a"  between  Ad'  and  this  curve  which 
will  be  proportional  to  the  thrust  of  the  earth  against  AB. 
This  maximum  distance  multiplied  by  the  altitude  Ac  -f-  2 
and  the  product  by  y,  the  weight  of  a  cubic  foot  of  earth, 
will  be  the  pressure  of  the  earth. 

This  method  is  perfectly  general  and  can  be  applied  in 
any  case. 

If  the  earth-pressure  is  assumed  to  have  the  direction 
given  by  the  formulas  of  the  preceding  theory,  the  con- 
struction will  give  the  same  value  of  E,  the  pressure  of  the 
earth. 

Some  writers  assume  that  the  direction  of  E  makes  the 
angle  0"  =  0  with  the  normal  to  the  back  of  the  wall  in 
all  cases.  This  assumption  cannot  be  correct  until  the  wall 
commences  to  tip  forward,  and  then  it  is  doubtful  that  such 
is  the  case  unless  the  earth  and  wall  are  perfectly  dry. 

To  be  on  the  side  of  safety  in  every  case,  it  is  better  to 
take  the  direction  of  E  as  given  by  the  above  theory. 

The  construction  of  Fig.  8a  will  give  the  maximum  thrust 
for  any  assumed  direction  for  any  case. 

TRAPEZOIDAL  WALLS. 

It  will  be  assumed  that  the  direction  and  magnitude  of 
the  earth-pressure  is  known,  that  the  position  and  extent 
of  the  back  of  the  wall  and  the  width  of  the  top  are  given, 


T8EORY  OF  EARTH-PRESSURE. 


49 


to  determine  the  width  of  the  base  for  stability  against  over- 
turning, sliding,  and  crushing  of  the  material. 


FIG.  9. 

Stability  against  Overturning.  —  Let  AB  CD,  Fig.  9,  rep- 
resent a  section  of  a  trapezoidal  wall,  TR  the  direction  of 
the  earth-thrust,  JG  the  vertical  passing  through  the  cen- 
tre of  gravity  of  the  wall,  and  JO  the  direction  of  the  re- 
sultant pressure  on  the  base  AD  caused  by  ^and  G. 

As  long  as  R  cuts  the  base  AD,  the  wall  will  be  stable 
against  overturning.  When  R  takes  the  direction  JQ,  the 
wall  may  be  said  to  be  on  the  point  of  overturning;  then 

ON 

the  factor  of  safety  against  overturning  is  ~~.,  where  ON 


is  the  actual  value  of  E,  and  QNthe  value  of  E  required  to 
make  the  resultant  R  pass  through  D. 

Stability  against  Sliding.  —  Since  the  wall  will  not  slide 


50 


RET AININO-W ALLS  FOR  EARTH. 


along  the  surface  DA  until  the  resultant  R  makes  an  angle 
with  the  normal  to  DA  greater  than  the  angle  of  friction 
0',  the  factor  of  safety  against  sliding  can  be  obtained  as 
follows:  Draw  JP  making  the  angle  JMU '=  0';  then 

PN 

the  factor  of  safety  against  sliding  is  -^,  where  PN  is  the 

force  required  in  the  direction  of  E  to  make  R  make  the 
angle  0'  with  the  normal  to  AD,  and  ON  the  actual  value 
of  K 

Stability  against  the  Crushing  of  the  Material. — In  ordi- 
nary practice  walls  for  retaining  earth  are  not  of  sufficient 
height  to  cause  very  large  pressures  at  their  bases,  but  it 
is  necessary  to  consider  the  subject  on  account  of  the  ten- 
dency of  the  bed-joints  to  open  under  certain  conditions. 


Let  AB,  Fig.  10,  represent  any  bed- joint  in  the  wall,  P 
the  vertical  resultant  pressure  upon  the  joint,  and  x0  the 
distance  of  the  point  of  application  from  the  centre  of  the 
joint. 

The  intensity  of  P  can  be  considered  as  composed  of  a 

p 

uniform  intensity  p0  =  j,  and  a  uniformly  varying  inten- 
sity p9',  so  ih&t  px=  p0  -\-  p».  Let  a  equal  the  tangent  of 
the  angle  CDE,  then  ;;/  —  ax  and  px  =  p0  -j-  ax. 


THEORY  OF  EARTH-PRESSURE.  51 

The  pressure  upon  a  surface  (dx)  —  the  joint  heing  con- 
sidered unity  in  the  dimension  normal  to  the  page  —  is 

pxdx  =  p0dx  -f-  axdx, 
and  the  moment  of  this  about  DB  is 
(p0dx  -f-  axdx)x. 

The  algebraic  sum  of  these  moments  for  values  of  x  be- 
tween the  limits  ±  —  must  equal  Px0,  or 


Px0  =        (p0xdx  +  ax*dx). 
Integrating, 


„        0       _ 
B3 


and 


or  - 


I2xx0 


and  if  x0  be  replaced  by  ^B  —  Q,  where  Q  is  the  distance 
from  A  to  the  point  where  P  cuts  the  base,  (Fig.  11,) 


and 


if  e=i5, 

p'  =  0    and 


52  RETA1NING-WALLS  FOR  EARTH. 

from  which  it  is  seen  that  when  R  cuts  the  base  outside 
the  middle  third,  the  joint  will  have  a  tendency  to  open  at 
points  which  are  at  a  maximum  distance  from  R  where  it 
cuts  the  base. 

Therefore  in  no  case  should  the  resultant  pressure  be 
permitted  to  cut  the  base  outside  the  middle  third.  This 
makes  it  unnecessary  to  consider  the  stability  against  over- 
turning. 


C     B      B 


Then  in  designing  a  wall  the  following  conditions  must 
exist  for  stability : 

I.  The  resultant  R  must  cut  the  base  for  stability  against 
overturning. 

II.  The  resultant  R  must  not  make  an  angle  with  the 
normal  to  the  base  of  the  wall  greater  than  the  angle  of  fric- 
tion 0'. 


THEORY  OF  EARTH-PRESSURE.  53 

III.  The  resultant  R  must  not  cut  the  base  outside  of 
the  middle  third,  in  order  that  there  may  be  no  tendency  for 
the  bed-joints  to  open. 

The  above  three  conditions  apply  to  any  bed-joint  of  the 
wall;  but  if  they  are  satisfied  at  the  base  and  the  wall  has 
the  section  shown  in  Fig.  11,  it  will  not  be  necessary  to 
consider  any  joints  above  the  base  unless  the  character  of 
the  stone  or  the  bonding  is  different. 

Determination  of  the  width  of  the  base  of  a  retaining- 
w  all  under  the  condition  that  R  cuts  the  base  at  a  point 
rom  the  toe  of  the  wall. 

Let  H,  B',  x,  d,  and  E  be  given  to  determine  B. 

From  Fig.  11, 

KF  —  -  sin  §  -{-  —  cos  8  --  —  sin  tf, 

DO  O 


_  -Bx-  2B'x  -  B" 


nv-nn    B  _B*  +  BB>-  Bx- 
"~ 


B') 

For  equilibrium 

E(KF)  =  G(HF)  =  E\B'  HW(HF). 

Substituting  the  values  of  A^and  HFin  the  above  and 
reducing,  it  becomes 


,    .     (8) 


54 


RETAINING -WALLS  FOR  EARTH. 


which  is  the  general  equation  for  the  width  of  the  base  of 
a  trapezoidal  wall. 

For  a  rectangular  wall  B'  =  B. 

For  a  triangular  wall  B'  —  0. 

For  a  wall  with  a  vertical  front  B'  -\-x-B  or 
B'  =  B  -  x. 

For  a  wall  with  a  vertical  back  x  =  0. 

Equation  (8)  is  easily  transformed  to  satisfy  the  require- 
ments of  special  cases. 

The  width  of  the  base  can  be  found  graphically  by  as- 
suming a  value  for  B  and  finding  the  value  of  Q-,  if  it  is 
less  than  %B  another  value  of  B  must  be  assumed,  and  so 
on  until  Q  is  equal  to  or  greater  than  ^B. 

Depth  of  Foundations. — Given  the  angle  of  repose  0  of 
any  earth,  to  determine  the  depth  to  which  it  is  necessary 
to  sink  a  foundation  to  support  a  given  load.  The  surface 
of  the  earth  is  assumed  to  be  horizontal. 


CASE  I.  When  the  intensity  of  the  pressure  on  the  base 
of  the  foundation  is  uniform. 

In  Fig.  12,  let  p0  represent  the  intensity  of  the  pressure 
on  the  base  of  the  foundation. 


THEORY  OF  EARTH-PRESSURE.  55 

Now  when  the  masonry  is  about  to  sink  (see  Eq.  (c)), 
p.       14-  sin  0  1  —  sin  0 

*      0      -  .  '  _  '  Q|«  Q      -      ftj          _  '_ 

q   '     1  —  sin  0  °  1  +  sin  0' 

If  x'  represents  the  depth  to  which  the  foundation  extends 
below  the  surface  of  the  earth  and  y  the  weight  of  a  cubic 
foot  of  earth,  then  yx'  equals  the  vertical  intensity  of  the 
earth-pressure  on  a  plane  at  the  depth  of  the  lowest  point 
of  the  foundation. 

When  the  wall  is  on  the  point  of  sinking,  the  earth  must 
be  on  the  point  of  rising,  or 

q     _  1  -|-  sin  0 
yx'       1  —  sin  0' 
or 


\±™L  +  r 

1  —  sin  0  ) 


In  any  case  p9  must  not  have-  a  greater  value  than  that  ob- 
tained from  (15)  — 


=  p. 

y         --  sin  y 


_  4>\ 

2J 


The  value  of  x'  as  obtained  from  (16)  is  the  least  allow- 
able value  consistent  with  equilibrium.  Since  x'  is  a  func- 

tion of  tan4  U5°—  |-j,  care  must  be  taken  that  0  is  assumed 

at  its  least  value.  As  0  becomes  smaller  the  value  of  x' 
increases  rapidly. 

CASE  II.  When  the  intensity  of  the  pressure  on  the  base 
is  uniformly  varying. 

Let  p  represent  the  maximum  intensity  of  the  pressure 
on  the  earth  and  p'  the  minimum  intensity;  then  for 


56  RETAINING  -WALLS  FOR  EARTH. 

equilibrium  p  must  not  exceed  the  value  obtained  from  the 
following  equation  : 


Also,  pr  must  never  be  less  than  x'y;  then 


7; -I-?/        Xr "V   ( 

TJ     =  '  '        —  '       \     l-|-(    ~      '     ^    I         \.    —    %'y     *-        I       "*•"•       V"  /I  Q\ 

2  2    (        U— sin0/    [          r  (1  — sin0)a> 

which  expresses  the  maximum  value  which  p0  can  have  for 
the  equilibrium  of  the  earth.     Solving  (18)  for  x', 

I        I       o^2~  ^/>   >  ....          (^/ 


which  is  the  minimum  value  x'  can  have  for  the  equilibrium 
of  the  earth. 

In  order  that  p9  may  never  be  less  than  x'y  the  result- 
ant pressure  on  the  base  of  the  foundation  must  cut  the 
base  within  a  certain  distance  of  the  centre  of  the  base.  If 
x0  equal  this  distance,  then  (see  page  51) 


Substituting  the  value  oipa  from  (18)  and  solving  for  x9, 

1       sin  0 


<2°> 


which  is  the  maximum  value  xa  can  have,  consistent  with 
the  stability  of  the  earth. 

Abutting  Power  of  Earth.  —  Let  the  surface  of  the  earth 
be  horizontal  and  the  body  pushing  the  earth  have  a  verti- 


THEORY  OF  EARTH-PRESSURE.  57 

cal  face;  then  at  the  depth  x'  the  maximum  horizontal 
pressure  per  unit  of  area  is  (see  Case  I  above) 

,     1  -f-  sin  0 


and  since  q  varies  directly  as  x',  the  total  thrust  P  which 
the  earth  is  capable  of  resisting  is 

_  (*')>  1  +  Bin  4>  m. 

5}      1  -  sin  0'     '    •     •     •     W 


APPENDIX. 


WEYRAUCH'S 
THEORY   OF    THE    RETAINING-WALL* 


Itf  the  following  the  earth  is  supposed  without  cohesion, 
and  its  pressure  is  determined  independently  of  any  arbi- 
trary assumptions  as  to  direction  of  the  earth-pressure,  and 
with  sole  reference  to  the  three  necessary  conditions  of 
equilibrium.  The  single  and  only  supposition,  then,  is  as 
follows:  That  the  forces  upon  any  imaginary  plane-section 
through  the  mass  of  earth  have  the  same  direction. 

This  assumption  lies  at  the  foundation  of  all  theories  of 
earth-pressure  against  retaining-walls.  For  those  cases, 
therefore,  to  which  the  following  discussion  does  not  apply 
no  complete  or  satisfactory  theory  is  yet  possible.  In 
what  follows,  the  ordinary  assumption  as  to  the  direction 
of  the  earth-pressure  will  be  proved  to  be  incorrect,  except 
for  special  cases. 

*  Zeitsclirift  fur  Baukunde,  Band  I.  Ilcft  2,  1878. 


60 


THEORY  OF  THE  RETAILING -WALL. 


GENERAL    RELATIONS. 

Let  the  surface  of  the  earth  have  any  form,  and  the 
wall  AB,  Fig.  1,  have  any  inclination.  The  earth-pres- 
sure makes  any  angle,  d,  with  the  normal  to  the  wall. 

Suppose  through  the  point  A  the  plane  AC.  Then  the 
weight  G  of  the  prism  ABC  is  held  in  equilibrium  by  the 


reaction  of  the  wall,  E,  arid  by  the  resultant,  R,  of  all  the 
forces  acting  upon  A  C. 

Now  decompose  E,  G,  and  R  into  components  parallel 
and  normal  to  AC;  then  for  every  unit  in  length  of  the 
wall,  denoting  by  e,  g,  and  r  the  lever-arms  of  E,  G,  and 
R  respectively  with  reference  to  A,  the  sum  of  the  forces 
parallel  to  A  C  =  0,  or 


P-P,_P  =0; 


(1) 


GENERAL  RELATIONS.  61 

the  sum  of  the  forces  perpendicular  to  A  0  =  0,  or 

Q  +  C.  -  <?,  =  o ; (3) 

the  sum  of  moments  about  ^4  =  0,  or 

Og  +  ^e  -  Rr  =  0 (3) 

Equation  (3)  was  first  introduced  by  Prof.  Weyrauch. 

Further,  according  to  the  theory  of  friction,  if  cp  is  the 
coefficient  of  friction  for  earth  on  earth, 

P   /  P  —  P    Z 

-^  ±  tan  <p  or  -Q^~Q   =  =  tan  V-      •    •  .  (4) 

If  now  there  is  any  plane  for  which 

P-Pl  =  (Q  +  Ql)imV,      ...     (5) 

P  —   P 

this  plane  A  C  will  be  a  plane  of  equilibrium,  and  77-7 


V  1^  1 

will  be  a  maximum,  or 

— ^£— = ° (6) 

This  plane  is  designated  as  the  "  surface  of  rupture." 
From  Fig.  1,  for  every  position  of  A-C, 

P  =  G  cos  GO,  Q  =  G  sin  c0, 

»4-«4-<n,      0,  =  J£cos(<»4-«4-(tt. 


Substituting  the    above  values  of  P,   Plf    Q,   and    Qv  in 
equation  (5),  it  becomes 


G  cos  co  —  ^sin  (G?  -}-  a  -\-  d) 

=  [^  sin  GO  +  ^cos  (68?  +  ^+  6)]  tan  ?>; 


62  THEORY  OF  THE  RETAINING -WALL. 

and  when  GO  refers  to  the  surface  of  rupture,  the  earth - 
pressure  upon  AB  becomes 

-p cos  GO  —  sin  GO  tan  cp 

~  sin  (GO  -\-a-\-  d)  -f-  cos  (GO-\-  a  -\-  6)  tan  cp 

Substituting  the  value  of  tan  cp  or -.  this  becomes 

cos  cp 

cos  cp  cos  GO  —  sin  GO  sin  cp 

sin  (GO  -j-  a  -f-  6)  cos  cp  -f-  cos  (GO  -j-  a  -}-  d)  sin  <£> 

which  becomes 

cos  (p  +  GO) 


In  order  to  refer  to  the  surface  of  rupture,  the  following 
relation  must  exist : 

,  ,'G  cos  GO  —  E  sin  (GO  -\-  a  -}- 


fG  c 

\6r  si 


sinaJ  +  J&ooB(<B+«41tf)i  =  0 


^ca 

Performing  the  differentiation  indicated  in  the  equation 
to),  considering  6r  and  <#  as  the  variables,  it  becomes 


+  [dG  COSO)  -  Sin  o>rfo>  »  -  J?7COS  (w  +  a+  8)da>l    [G  Sin  w  4  ^COS  («+  a+  «)] 

—  [d(r  sin  a>  +  cos  a><ia>  G  —  E  sin  (a>  4-  a  +  8)da)]   [G  cos  a>  -  -g  sin  (a>  -\-  a  4-  5)]    _ 

[G  sin  a>  +  £  COS  (<o  '+  a  +  6)]2rfw 
-0;      ..................    .    ........    (76) 


dividing  by  dco,  this  becomes 


[G  sin  u>+  ^  COS(a>  +  a  +  5)] 


r  dg  sin  a>  cos  w  -i7  sin  (»+  a-f  5)]]  [G  cos  co  -  #sin  (»  +  a  +  5)] 

~~  L         aw 


[&  sin  w  +  E  cos  (w  +  a  4-  6)J2 

......   (re) 


GENERAL  RELATIONS.  63 

or 


d£coscor^, .....       .„___,     ,       •  M_[<ysin 


-1*5-^  [<?  COS  to  -  Esin  (co  +  a  +  6)]  -  [Q  COS  co  - 


[G  sin  co  -|-  £;  cos  (w  +  a  +  8)]a 
=  0  .............................    (7d 

Now,  since 

cos  GJCOS  ((&-{-  a  -{-$)  -\-sin  GO  sin  (oj-\-a-\-d)=  cos  (« 
and  sina  GL>  +  cos2  GL)  —  1, 

by  clearing  of  fractions  this  becomes 

og(«+j)  +  <y_8g  +^  =  ft( 

wCe? 

Now  since  ^6r  =  J^  .  JG^  .  Icy,  equation  (7e)  reduces  to 


(T  -2GE  sin  («  +rf)  -  +  J5»  =  o, 

A 


which  becomes,  after  dividing  by  GE, 


G  ^Vcos(«+d)   ,   E 

,-2  sin  (or  +  d)  -          ~~         +^ 


•pi 

Substituting  the  value  of  -^  from  equation  (7),  transposing 

and  multiplying  by  two,  equation  (8)  reduces  to 

a>  +  6)  _  2  cos  (.ft  +  to)          ^Y^^_+I),  (8a) 

cos  (^  +  w)  "1~" 


64  THEORY  OF  THE  RETAINING  -WALL. 

whence 


which  reduces  to 

0  _  __  cos  (<t>  -r  to)  sin  (0  -f  <o  4  a  -f  S)  cos  (a  -f  S)fc2y  ,g  x 

2  [sin*  (</>-|-<o-|-a+S)-2  sin  (a-f  S)cos  (4>+w)  sin  (<J>  -f  w+a+S>fcos2  (</>+«)]  ' 

Since 

sin  ((p  +  c»  4~  «  +  tf)  —  sin  (^  +  ^  cos  (a  +  ^) 
+  cos  (9?  +  ct>)  sin  (a  +  d), 

the  parenthetical  portion  of  the  denominator  becomes 


-f  2  sin  (<*+#)  cos  ((p+Go)  sin  (^4-^)  cos 
+  cos2  (^?+(»)  sin2  (a+6) 

—  2sin(a4-(J)  cos  (^+c^)  sin 

—  2  sin   ^(5'  cos        -^  cos 


or 


cos2  (or+tf) 

—  2  sin2  (tf+tf)  cos2 


or        sin'  (<p-\-  &>)  cos2  («+^)  —  cos3  (q>-{-Go)  sin2  (« 

+  COS2  (<p+GO), 


cos2  (a+6)  +  cos2  (9>+<»)  [1-sin2  (a+3)], 
?)  cos3  («-f 
or  cos*  («+ov)  [sin2 


GENERAL  RELATIONS. 


65 


which  equals  cos2  (<x-\-d),  and  equation  (Sc)  becomes,  after 
dividing  by  cos  («+#)  and  factoring, 


G  =  - 
from  which 


o)  sin  (<p-\-Go-\-a-}-6)     l?y      ^       ,. 
—  TAT-^—    —  J  —  -  •  -IT"  =  Function  y,  (9) 
cos(ar+#)  2 


sn 


cos 


which  being  substituted  in  equation  (7)  gives 

<7cos((?-fc3)  _  cos2  (<?+&?)      fry 

O  />  ,-,,rw,  /  x,,    I    *£\  «^o     /  ^/    I    x*\  O     *       *       V        / 


cos 


FIG.  2. 


And,  since  the  sum  of  the  horizontal  components  of  E,  G, 
and  R  must  be  equal  to  0,  or  Fig.  2, 


R  cos  (GO-\-  cp\ 


and 


~  Jit 


66 


THEORY  OF  THE  RETAINING-WALL. 


which   becomes,  after  substituting  the  value  of  E  from 
equation  (1.0), 


.  .       (11) 


Let  AD,  Fig.  2,  be  the  natural  slope  of  the  ground. 
From  C  let  fall  the  perpendicular  CH,  and  draw  CJ,  mak- 
ing the  angle  (<*-}-#)  with  CH  ;  then 


cos 


FIG.  2. 


The  expression  for  AJ  is  obtained  in  the  following  man 
ner  (Fig.  2): 

CH—  k  cos  (q>  +(*)),      AH—  &  sin  (<p-\-Go), 
HJ  :  CH  ::  sin   tf+tf    :  cos  ( 


in  (o-j-tf)  _  cos(^+cj)sin  (0^7 
"  ' 


Tl+HJ=AJ 

sin     )  +  ft?   cos 


cos 


cos  (cx-^-o) 


GENERAL  RELATIONS.  67 

which  reduces  to 


•**•**   —  -  /  -  1  —  j>\  --  K  I 

cos 


and  hence,  according  to  equation  (9), 


Also,  if  ^4^Tis  perpendicular  to  CJ, 

CH  £cos    <GO  E 


AK  ~  k  sin  (cp-\-co-\-a-\-6)  ~  G  9 

and  if  JL  is  made  equal  to  JC,  then,  since  the  perpendicu- 
lar from  L  upon  CJ  is  equal  to  CH, 

ACJL  _  CH    _E 
ACJA~  AK~  G' 

or  E=yACJL  ......     (13) 

If,  finally,  AM  =  AC, 

AACM=  AM'CIL  =  j  ^  cos  (p+(»), 
A 

or  R  =  yAACM.      .....     (14) 

All  these  geometrical  results  may  be  summed  up  as  fol- 
lows : 

Draw  from  the  highest  point  C  of  the  surface  of  rupture 
a  line  CJ,  which  makes  with  the  normal  CH  to  the  natu- 
ral slope  the  angle  a  +  d,  or  the  angle  which  the  earth- 
pressure  makes  with  the  horizontal  •  then  the  AACJ  is 


68      THEORY  OF  THE  RETAINING- WALL. 

equal  in  area  to  the  A  ABC,  the  prism  of  rupture.  Then 
lay  off  JL  =  JG  and  AM=  AC  and  draw  CL  and  CM ; 
then  for  every  unit  in  length  of  the  wall  the  following 
relations  exist : 

Weight  of  prism  of  rupture,  G  =  yACAJ\    j 

Earth-pressure  upon  wall,  E  —  yACJL\    \  (14«) 

Reaction  of  the  surface  of  rupture,  R  =  yACAM.  ) 

The  first  two  relations  were  first  made  known  by  Rebhahn 
in  1871,  for  d  =  0  or  cp. 

Since,  now,    G  :  E  :  R  =  AJ  :  JC  :  CA,     .     .     .     (15) 

it  can  be  asserted  that — 

The  weight  of  the  prism  of  rupture  and  the  reactions  of 
the  wall  and  of  the  surface  of  rupture  are  to  each  other  as 
the  three  sides  of  the  AACJ. 

Thus  far  no  assumption  whatever  has  been  made  as  to 
the  value  of  the  angle  d.  This  is  determined  by  equation 
(3),  which,  in  all  theories  following  Coulomb's  method, 
does  not  occur. 


PLANE  EARTH-SURFACE  INCLINED. 


II. 


PLANE  EARTH-SURFACE  INCLINED 

ADOPT  in  this  case  the  notation  of  Fig.  3,  and  let  E  be 
first  determined  for  any  value  of  S. 


FIG.  3. 


If  A  C  is  the  surface  of  rupture,  then  A  ABC '  =  AACJ\ 
or,  since 


__sin    II 
277  ~  shTlTP 


In  like  manner,        AJ  =  A  C 


AB  =  AC 


sin    V 


sin   II 
sin  III  * 


or 


sin  vr 

But  since  A  ABC  =  A  A  CJ, 

AB  .  A  <7sin  I  =  AJ .  A  C  sin  /F; 

sin  /sin  //_  sin  IV sin  V 
sin  IlT~  sin  VI      '    * 


(1C) 


70 


THEORY  off  THE 


or,  finally, 
sin  (<*-}-&?)  cos 

=  sin 


cos  (a-\-d) 
cos 


cos 


Further,  from  Fig.  3,  if  BN\&  perpendicular  to  AD, 

AADB  =  24AJC+4JDC, 
or  AD  .  BN=  2AJ  .  CH+JD  .  CH\ 


and  since 
and 


BN      BO       OD 


CH  ~     CJ        JD  ' 


AD.  OD  =JD(AJ+AD], 
AD  (AD- A  0)  =  (AD-AJ)(AJ+AD), 
whence  AO-._  AJ  =  AJ .  AD.      .     .     .       (1?) 


FIG.  3'. 


Upon  this  relation  rests  the  well-known  construction  of 
Poncelet  for  the  earth-pressure,,  Draw  (Fig.  3;)  BN  per- 
pendicular to  the  natural  slope  AD\  draw  BO,  making  the 
same  angle  with  BN  that  E  makes  with  the  horizontal,  and 


PLANS!  EARTH-SURFACE  INCLINED.  71 

then  determine  the  point /so  that  equation  (17)  is  ful- 
filled, that  is,  make  AJ  a  mean  proportional  between  A  0 
and  AD  ;  then  draw  JC parallel  to  OB.  Thus  the  surface 
of  rupture  AC  is  found,  and  use  can  now  be  made  of  the 
relations  already  deduced  in  I. 

In  order  to  determine  J  (A,  0,  and  D  being  given),  there 
are  several  methods,  one  of  which  is  indicated  in  the 
figure.  In  all  these  constructions  d  is  assumed. 

Now  from  equation  (13),     E  =  |  y  JC    cos  (a  -}-  d), 
but 


A/~AO 
1-  V  - 


CJ  =  AD-  AJ  _  AD-  V  AD.  AO  _   x  ~  r    AD 
BO~AD~-AO~  AD-AO  ~A~0 

~AD 

Let      n  =  i/?          then     CJ  =    = 


—  ft  n 

From  Fig.  3, 

AO  _  sin  (cp+  d)  AB  _  sin  (cp  —  e) 

AB  ~  cos  (a  +  6)'         ~AD  .     cos  (a  —  "7); 

and  the  multiplication  of  these  equations  gives 

./sin  (y  -R)  sin  (y  -  g) 
cos  (a  +  6)  cos  (a  -  £)' 


72  THEORY  OF  THE  RflTAlNXNG-WALL. 

and  by  substitution  of  BO  and  n  in  the  value  for  CJ,  and 
of  CJ  in  that  for  E, 


I-  COS  (.*,-  a)  H»  _  Py_          r    COS(^-a)  -]«         fe'y 

L      n-fl       J  2  cos  (a  4-  5)      L(n-fl)  cosaJ  2cos(a+*)'    ' 

For  the  special  case  of  the  earth-surface  parallel  to  the 
angle  of  repose,  e  =  cp,  n  =  0,  and 

cosa  (<?-<*)  Fy       rcos(tp—a)-\*        l?y 
cos(a+6)     2    '     [_      cosa~   J  2  cos  (<x+d)'(' 

Those  formulae  hold  good  for  any  value  of  d.  But  the 
angle  &  is  determined  by  equation  (3).  In  order  to  insert 
e  and  r  in  this  formula,  the  points  of  application  of  E  and 
R  must  be  known.  The  angles  d  and  GO  are  connected  by 
the  relations  in  (16&),  in  which  there  are  no  other  unknown 
quantities.  Since  now  d,  according  to  the  single  assump- 
tion of  Prof.  Weyrauch's  theory,  is  independent  of  the 
height,  so  also  is  GO,  and  then  for  variable  li  equations  (19) 
and  (11)  become 

E  =  or,  R  =  cjt, 


Let  x  and  z  equal  the  distance  of  the  point  of  application 
of  E  and  R  from  A,  respectively.  Now  considering  the 
top  as  the  origin  or  centre  of  moments, 

E(l-x)  =  ZC  fjdl,          R(k-z)  =  Z  C^tfdk, 

and  therefore  x  =  $1    and    z  =  $k. 

Now  G  must  act  through  the  centre  of  gravity  of  the 
A  ABC,  and  it  has  been  already  proved  that  the  points 


PLANE  EARTS-&IJRFACE  INCLINED.         13 

of  application  of  E  and  R  are  at  distances  JZ  and  ^Tc  re- 
spectively above  A-,  hence  (Fig.  3')  ah  =  e<2  and  hf=g  = 
M  —  ah  =  %Jc  sin  GO  —  $1  sin  or. 

Substituting  these  values  in  equation  (3)  and  referring  to 
equation  (15), 

AB(CJcosS-  A  J  sin  a)  =  AC  (AC  cost  -AJsin<*\     .    .    .      (22) 

or 

sin  //(sin  /Fcos  8  -  sin  Fsin  a)  =  sin  ///(sin  F/cos  <£  -  sin  Fsin  w),  (22a) 
or  cos  (e  -f-  a>)  [cos  (<f>  +  w)  cos  6  —  sin  ($  +t>>  +  a  +  S)  sin  a] 

=  cos  (a  —  e)  [cos  (a  -f  6)  cos  <£  -  sin  (</>  -f-  w  +  a  -j-  5)  sin  w].   .    .    (226) 


By  means  of  the  two  equations  (16#)  and  (22b)  the  two 
unknown  quantities  d  and  &)  are  completely  determined. 
As  soon  as  these  are  known,  ,Z?can  be  found  from  equation 
(19)  or  (20).  Also  by  the  relations  in  equations  (16)  and 
(22),  or  (\6a)  and  (22Z>),  the  surface  of  rupture  and  direc- 
tion of  the  earth  -pressure  may  be  determined,  and  can 
therefore  be  found  by  a  graphical  construction. 


THEORY  OF  THE  RETAINING- WALL. 


III. 
HORIZONTAL  EARTH-SURFACE. 

FOR  this  most  important  practical  case  it  is  simply  nec- 
essary to  make  £  =  0  in  equation  (19).  The  proper  values 
of  d  and  GO  in  this  case  are  found  from  (16£)  and  (221). 

Making  €  =  0  in  equation  (22b),  it  becomes 

cos  GO  [cos  (cp  -{-GO)  cos  d  —  sin  (cp  -(-  GO  ~\-  a  -f  #)  sin  a~\ 

-  cos  a  [cos  (a  -\-  d)  cos  cp  —  sin  (cp-}-  GO-\-  a-\-  d)&\\\Go]  —  Q. 

Since 

sin  (cp-}-Go-{-  a-\-  d)  =  sin  (<p  +  ^  cos  (**  +  ^) 

+  cos  (cp  -|-  c^)  sin  (a  -j-  d), 
cos  (a  -f-  (J)  =  cos  tf  cos  (^  —  sin  a  sin  tf, 
and          sin  (a  -j-  tf)  —  sin  or  cos  d  -j-  cos  a  sin  tf, 

the  above  expression  becomes 

cos  GO  cos  d  cos  (<£>  +  GL>) 

—  cos  GO  sin  a  cos  acos  d  sin  (9?  -f-  GO) 

«-]-  cos  G?  sin2  a  sin  #  sin  (cp  -f-  6?) 

—  cos  GO  sin  or  cos  <*  sin  6  cos  (<p  -j-  GO) 

—  cos  G?  sin2  a  cos  tf  cos  (<p  -f  GO) 

-  cos  «  cos  q>  cos  (or  +  6) 

-}-  cos2  «  sin  6?  cos  #sin  (cp  -{•  GO) 

—  cos  a  sin  GO  sin  #  sin  d  sin  (<^  -j-  GO) 
+  cos2  «  sin  a?  sin  d  cos  (^?  -f-  GO) 

-|-cos  ^  sin  GO  sin  #  cos  d  cos  (cp  -j-  G?) 


HORIZONTAL  EARTH-SURFACE.  75 

fr 

which  reduces  to 

COS  ft?  COS  (<£>+  GO)  COS  S  ~] 

—  sin  a  cos  a  [sin  (<p-{-  GO)  cos  Go  —  cos(cp-{-  GO)  sin  GO]  cos  tf  I 

—  sin  tfcos  or  [cos(<^-f~G!:7)COSft:?~h  sin(<^-|-  c^sinG?]  sin  6  \ 
-{-[sin^sin  (cp  -f-  GO)  cos  c^-f-  cos2<*  cos  (cp-\-co)  sin  02]  sin  6  I 
-j-  [cos2 a  sin  (<p  +  G?)  sin  00  —  sin2<*cos(<p-|-G0)cos  a?]  cos  tf  | 

—  cos2**  cos  cp  cos  d  -j-  sin  a:  cos  a  cos  <p  sin  d 

=  0 (22c) 

The  expression  in  the  first  parenthesis  is  equal  to  sin  cp, 
in  the  second  to  cos  cp.  If  in  the  third  cos2  a  =  1  —  sin2  a, 
and  in  the  fourth  sin2  a  —  1  —  cos2  a,  equation  (2%c)  be- 
comes 

+  cos  GO  cos  (cp  -j-  CL>)  cos  #  —  sin  a  cos  «  cos  $  sin  cp 

—  sin  a  cos  a  sin  #  cos  cp 

-j-  sin  (J  sin2^  sin  (cp-\-oo)  cos  c^-f-sin  tf  sin  c» cos  (cp-\-co) 
—  sin2**  sin  GO  sin  d  cos  (cp  -j-  &?) 

-j-  cos  d  cos2ar  sin  (<£>-]- GS?)  sin  GO  —  cos  (Jcosc^cos  (^-j-^) 
+  cos2  a  cos  (^  cos  GO  cos  (<7?  -j-  GO) 

-  cosV  cos  9?  cos  d"  -f-  sin  a  cos  a  cos  <p  sin  d 

Keducing  and  dividing  by  cos  #, 

-  sin  a  cos  a  sin  <p  +  sin2  ^  c°s  G?  sin  (cp  -j-  cw)  tan  d 

-f-  sin  GO  cos  (<p  -}-  GO)  tan  ^ 

-  sin2  a  sin  G?  cos  (cp  -\-  GO)  tan  # 

-I-  cos2  a  sin  G?  sin  (cp  -f-  cy) 

+  COS3  <*  COS  Cz5  COS  (cp  -j-  &?)  —  COS2  a  COS  <p 


Since 


cos  GO  sin  (9?  -j-  G?)  —  sin  GO  cos  (<p  -j-  GO)  =  sin 
2 


76  THEORY  OF  THE  RETAINING-WALL. 

and 

sin  co  sin  (cp  -f-  GO)  -J-  cos  <#  cos  (9?  +  ca)  =  cos  <p, 
this  reduces  to 

—  sin  a  cos  «  sin  cp  -\-  sin2  «  sin  cp  tan  # 
-f  sin  GO  cos  (9?  +  GO)  tan  #  =  0; 

and  since 

cos  (cp  -{-  GO)  sin  c^  =  ^  sin  (2oo  -j-  ^>)  —  |  sin  ^>, 

this  becomes 

_  2sin  <r  cos  <y  sin  cp       __  ^ 

~  2sin2  a  sin  <p  +  sin  (%GO  -\-  cp)  —  sin  <p  ' 

and  since 

sin  a  cos  a  —  %  sin  2a     and     1—2  sin2  a  =  cos  2a, 

this  reduces  to 

,      ^  _  sin  <    sin 


sn    2otf  -|-  ^?  —  sn  cp  cos    a 

This  equation,   therefore,   expresses  the  condition  that 
the  "  sum  of  the  moments  of  E,  G,  and  R  is  zero." 

Substituting  ^-5  for  tan  d  in  equation  (23),  clearing 
cos  o 

of  fractions  and  factoring, 

sin  d  sin  (2ft?  +  cp)  —  sin  d  sin  cp  cos  2a  =  sin  cp  cos  <5  sin  2tf, 


HORIZONTAL  EARTH-SURFACE.  77 

or 

sin  d  sin  (%co  -\-  cp)  =  sin  cp  cos  d  sin  2or  -J-  sin  <p  sin  6X  cos  2or. 

Since      cos  #  sin  2a  -j-  sin  #  cos  2a  =  sin  (2or  +  6"), 
this  becomes 

sin  d  sin  (2oo  -{-  cp)  =  sin  cp  sin  (2<*  +  #).     .     (24) 

In  order  to  determine  GO  it  is  only  necessary  to  make  £  =  0 
in  equation  (16Z>)  express  sin  (cp  -j-  ca  -f-  a  -j-  o")  in  terms  of 
sin  and  cos  (cp  -f  GL>)  and  (or  +o'),  and  then  the  sin  and  cos 
of  (a  -f-  6)  in  terms  of  the  sin  and  cos^of  a  and  #.  Mak- 
ing s  =  0  in  equation  (16#),  it  becomes 

sin  (a  -f-  G?)  cos  (a  -f  tf)  cos  G? 

—  sin  (cp  +  GL>  -j-  a  +  6")  [cos  (9*  +  a?)  cos  a].    (24«) 

Since 

sin  (cp  -\-  GO  -\-a  -f-  3)  —  sin  (^?  -f-  c«?)  cos  (a'  -|-  6) 

4-  cos  (^?  -j-  a?)  sin  (a  -|-  °") 
sin  (a  -+-  d)  =  sin  a  cos  d  -f~  cos  a'  sin  # 
cos  (or  +  d)  =  cos  a  cos  d  —  sin  a  sin  #; 

hence 

sin  (cp  -\-  GO  +  a  +  #)  =  sin  (cp  -\-  GO)  cos  a  cos  # 

—  sin  (^  -j-  GO)  sin  <*  sin  d 
4-  cos  (^  +  GO)  sin  <*  cos  d 

-j-  cos  (^>  4~  ^  cos 


78 


THEORY  OF  THE  RETAINING-WALL. 


and  equation  (24«)  reduces  to 

cos  oo  sin  (a  -\-GO)  cos  a  cos  6  ^ 

—  cos  oo  sin  (a  -j-  GO)  sin  a  sin  d 
—  cos2  a  cos  (cp  -4-  GO)  sin  (<z>  -4-  GO]  cos  # 

V    _   f\     /  f\  t  -j  \ 

-f  cos  <x  cos  (cp  -|-  &?)  sin  (<p  -f-  &?)  sin  a  sin  d  \ 


—  cos  <*  cos 


6?  sn 
+  GO)  sin 


"  =  0.  (24c) 


Dividing  by  cos  #, 

cos  a  cos  GO  sin  (#  -f-  GO) 

—  cos  GO  sin  a  sin  (or  -j-  &?)  tan  (5" 

—  cos2  #  cos  (cp  -f-  G?)  sin  (<p  -j-  GO) 

-\-  cos  «  sin  a  cos  (^?  -j-  GO)  sin  (<p  -|~  GZ?)  tan  $ 
-  cos  <*  sin  a  cos2  (9*  -j-  ^0 

—  cos2  a:  cos2  (cp  -\-  GO)  tan  d 

Since 


cos  a'  cos  GO  sin  (or  -f  GO)  equals,  by  expanding  sin  (a  -\-  GO), 
sin  a  cos  a  cos2  G£>  -f-  sin  GO  cos  G£>  cos2  a,  and  likewise 

—  cos  GO  sin  <*  sin  (<*  -f-  GO)  tan  #  =  —  cos2  GO  sin2  <*  tan  d 
-  cos  or  sin  a  cos  ca  sin  GO  tan  #, 

equation  (24c)  becomes 


—  sin  a  cos  a  [cos2  (<p  -f  &?)  —  cos2  GL>] 
—cos2  a  [sin  (cp-\-co)  cos  (^>  +  <*>)  —  sin  G!?COS  c^ 

—  [cos2  a  cos2  (^  -f  GO)  -f-  sin2  ^  cos2  &?]  tan  d 
-)-  sin  a  cos  «  [sin  (cp  -j-  GJ)  cos  (<^  -f-  GJ) 

r-  sin  cy  cos  GO]  tan  ^ , 


^  =  0. 


HORIZONTAL  EARTH-SURFACE. 


79 


Now 


COS2  (q)  -\-  GO)  —  COS2  GO  = 


cos2(<z>4-  GO)—  cos  2  GO 

*    /Vl    V    '  ' ' 


which  equals 

2  sin  \  [Zoo  — 


sn 


_  2  sin  (—  <p)  sin 


or 
and 


—  sin  (2o?  -f-  ^)  sin  9?, 


sin  (cp  -f-  GO)  cos  ((p  -f-  GO)  —  sin  GO  cos  &? 

=  -J  sin  2((p  -\-  GO)  — 

also, 

sin  %a                           cos  2^ 
sin  a  cos  «  =  — - — ,  and  cos  a  — — 


sn 


Hence,  after  multiplying  by  2,  equation  (24:d)  reduces  to 


sin  2<x  sin  (2c!?  -j-  (p)  sin  9? 

—  cos  2a  -J  sin  2(<^>  -f-  G^)  -j-  cos  2«  £  sin  2 
—  Y  sin  2(<p  -|-  GZ?)  -f-  -J-  sin  2c«9 

)— cos2  (^-J-G?)  tan 


—  tan  #  cos  2  or  cos2 


-  2  tan  #  sin2  a  cos2  GO 

-f-  sin  2or  sin  (q>  -\-  GO)  cos  (cp  -J-  o?)  tan 
—  sin  %<x  sin  c»  cos  GO  tan  £ 


80  THEORY  OF  THE  RETAINING  WALL. 

Now 

—  2  tan  6  sin3  a  cos2  GO  =  [since  sin2  a  =  1  —  cos2  a] 

—  [cos2  a?  —  cos2  a  cos2  ctf]  2  tan 

which  equals 

—  2  cos2  &3  tan  d  -J-  2  tan  d  cos2  a  cos2  to. 
Also, 


cos  2a  sin  2(<p  -j-  <**)     •   cos 

~~ 


r  sin  2(<#  4-  GJ)  —  sin 

cos  2«  L—  ^jy- 

cos  2«[2  sin  cp  cos  (2o?  +  ^)] 


>~| 

j 


=  —  cos  2<y  cos  (2&?  -(-  y)  sin  <y?, 
and 

•  sin 


-}-  GO)       sin  So?  _       sin  2(q>  -\-  GO)  —  sin 

"  ~T~"  ~2~ 


_  2  sin  j  (2<p  +  2&9  —  2co)  cos  j  (2<^?+  2&j  -f  2  GO) 

2 
=  —  sin  cp  cos  (2co-t-  ^)> 

and 

—  tan  d  cos  2a  cos2  (^?  +  GL>)  -f-  2  tan  #  cos2  a  cos8 

/.          .  .  cos  2a         \ 

=  (by  making  cos  a  =    —  -  ---  1-  ^  1 

-  tan  6  cos  2a  [cos2  ((p  +  GL>)  —  cos2  &?]  +  tan  ^  cos2 
pr       tan  61  cos  2a  sin  (2&7  4-  q>)  sin  ^  -j-  tan  #  cos2  G?; 


HORIZONTAL  EARTH-SURFACE. 


81 


—  tan  d  sin 


[• 


Also,  -  cos2  (cp  -f-  GO)  tan  d  -f-  tan  $  cos2  GO 

=  —  tan  d  [cos'"  (cp  -}-  GO)  —  cos2  GO] 
=  sin  <p  pin  (2&?  -f  ff>)  tan  #. 

Also, 

tan  d  sin  2«  sin  (cp  +  G?)  cos  (cp  +  GL>) 
—  sin  2<*  sin  GI>  cos  GJ  tan  # 

=  tan  d  sin  2a  [sin  (9?  -f"  *«*)  cos  (^  ~i~  ^  ~gin  ^  cos 
sin  2(^>  +  GO)  —  sin  2&9~| 

-      a  ~J 

=  tan  #  sin  2(Y  sin  <p  cos  (2fc>  +  <^>); 
and  hence  equation  (240)  becomes 
+  sin<p[sin(2a 

+  sin  cp  [sin  (2 


—  cos 
—  sin  9?  cos  (2&)  + 
cos  2a 


=0,(24/) 


cos  (2  G? 


+  sin  cp  [sin 
and 


sin  2or]  tan 
L)  tan  # 


sin  </>  [sin  (Su  +  4>)  sin  2a  -  cos  (2a>  +  ft)  cos  2a]  -  sin  <f>  cos  (2a>  +  <fr) 
2  cos2  a>-  sin  <f>  [sin  (2w+<£)  cos  Sa+cos  (2w+<A)  sin  2o]  -  sin  </>  sin  (2w+</))' 


By  making  sin  2o:  =  2  sin  ar  cos  a  and  cos  2«  =  1  —  2  sin2  « 
in  the  numerator,  and  cos  2a  =  2  cos  a  cos  a—  1  and  sin 
2tf  =  2  sin  #  cos  a  in  the  denominator,  this  becomes 

% 

tan  5  = 

sin  $  [sin  (2«-H>)  2  sin  a  cos  a—  cos  (2to4<fr)  +  cos  f&j-f  (fr)  2sin2  a]  —  sin  <fr  cos  (2w-H>) 

2cos2w-sin^[sin(2w-H>)2cos2a-sin(2w+«/.)+cos(2w+</»)2sina  cosa]-si 


or 


_ 
tanS= 


2  sin  <f>  sin  a  [sin  (2to+«ft)  cos  a+cos  (2<o+<^)  sin  a]  -  2  sin  <fr  cos  (2a)+</>) 


82  THEORY  OF  THE  RETAIN  ING-WALL. 

which  reduces  to 

tan  d  = 

sin  cp  sin  a  sin  (%GJ  -f-  cp  -\-  oi)  —  sin  cp  cos  (2&)  -f-  cp) 
cos2  oo  —  sin  cp  cos  a  sin  (2&)  -\-  cp  -f  a) 


Equating  this  value  of  tan  d  with  that  in  equation  ('23), 
sin  cp  sin  a  sin  (2a)  -\-  cp  -\-  a)  —  sin  cp  cos  (2oo  -f-  9^) 


cos"  &?  —  sn  ^  cos  a  sn 
sin  <£>  sin  2a 


-j-  <^>  -}- 


~  sin  (2ct?  -j-  cp)  —  sin  90  cos  2a  ' 

Dividing  by  sin  cp,  clearing  of  fractions  and   dividing  by 
sin  a,  also  transposing,  this  becomes 

sin  (2cj  -[-  cp  -f  a)  sin  (2&J  -f-  <p)  ^ 

—  sin  (2oo  -\-cp-\-oi]  sin  e?  cos  2a  —  -  -----  cos2  02 

sin  «: 


.    sin  2a  x    . 

—  cos  #  sin  (;i<i?  -\-cp-\-a)  sin 

cos  (^&9  -f-  cp)  [sin  (2c»  -f  cp]  —  sin 
sin  a 


=  0, 


or 


sin  (2&?  -j-  9?  -f  ^)  sin  (2cy  -}-  cp) 
—  sin  <T?  cos  2<^  sin  (2c^  -|-  cp  -\-a)  —  2  cos  a  cos 
-f  sin  cp  2  cos2  <*  sin  (2oo  -}-  cp  +  a) 

cos  (2ft?  -f-  cp}  [sin  (2(i9+  ^>)  —  sin  cp  cos  2or] 
sin  a 


J 


Since 


2  cos2  «f  —  cos  2<r  —  1  , 


HORIZONTAL  EARTH-SURFACE.  83 

this  becomes 

sin  (2oj  +  tp  +  a)  [sin  (2&?  +  9?)  -f-  sin  <p] 

-  2  cos  <*  cosa  a?  —  D  =  0, 

in  which 

n   _  cos  (260  -|-  <p)  [sin  (2ft?  -f  (p)  —  sin  cp  cos  2«] 

X/   —   ----  ;  . 

sm  or 
or 

sin  (2G?-f-^-f  a)  [2  sin  (<£-{-<p)  cos  GL>]  —  2  cos  a  cosac0—  /)=  0, 
or 

—cos  a  cos  cy—  -  --  = 


The  formulae  for  &?,  d,  and  E  can  now  be  found  in  the 
simplest  manner.  Equation  (25)  is  satisfied  for2co-\-(p=  90°. 
Hence, 


Substituting  this  value  in  equation  (23),  it  becomes 

sin  cp  sin  2a 


tan  #  = 


sin  (90—  <p-\-q>]  —  sin  cp  cos  2« 

(27) 


1  —  sin  ^?  cos  2«' 

or  the  equivalent,  but  more  convenient  expression  for  cal- 
culation, 

.     .     .     .     (28) 


84  THEORY  OF  THE  RETAINING-WALL. 

If,  finally,  GO  =  45°  —  ^  in  equation  (10),  it  becomes,  re« 

72 

membering  that  k*  =  — — , 
cos'cy 


E  = 


3  008' (45°- |) 


cos 


'  (45°  -  1 


hence     ^  tan' 

or,  from  equation  (28), 
.„  tan  a 


This  last  expression,  however,  when  a  =  0  takes  the  in- 
determinate form  —  . 

The  earth-pressure  upon  a  portion  of  the  wall  reaching 
from  the  depth  A0  to  the  depth  H  =  A0  -{-  A,  may  be  found 


HORIZONTAL  EARTH-SURFACE.  85 

from  equation  (29)  by  substituting  H*  —  h0*  in  place  of  If, 
as  is  evident  from  the  following: 

7T2 

Suppose  the  wall  to  have  a  height  H,  then  E0  =  U*^rY> 
and  likewise  for  a  height  h0 

El  =  C0  —y  .  • .  E  =  EQ  —  E\  =  CQ — -  y,  .     .  (29&) 

C0  representing  the  constant  quantity. 

From  equation  (29£)  E  =  C(H*  —  7*02);  hence  dE  = 
2  CHdH  —  2  Ch0dk0.  Now  let  x  equal  the  distance  of  the 
centre  of  pressure  below  the  top  of  the  wall,  then 

Ex=2C  fH  H*dH-  2C   Phfdh, 

V  Q  C     0 

or  C(H>  -  h*)x  =  %CH*  -  f  6V, 

2  tr  -  n: 

or  x  —  ~*  ~fn — y^j 

and  if  y  =  the  distance  from  bottom, 

^    -,,f^-     •     •     •     •      (3°) 

Equation  (30)  holds  good  when  the  earth-surface  is 
loaded  and  the  loading  is  equal  to  a  distributed  load  of  the 

height  h0.     Still,  even  then,  h0  is  often  so  small  that  —  can 

o 

be  substituted  for  it  just  as  for  unloaded  earth-surface.. 
In  all  cases  #  is  determined  by  equation  (28). 


86  THEORY  OF  THE  RETAINING- WALL. 

Instead  of  using  equations  (28)  and  (29),  the  following 
simple  construction  can  be  used  : 


FIG.  4. 

Draw  (Fig.  4)  AC  and  A  D  vertically  and  horizontally, 
each  equal  to  h,  also  DP  making  the  angle  FDG  =  45°  -  -  *| 

with  the  horizontal.  Through  the  points  D  and  ^describe 
a  circle  whose  centre  lies  in  AD.  Then  draw  GH  parallel 
to  AB,  and  through  A  the  straight  line  HJ.  Then  JG  is 
the  direction  of  the  earth-pressure  upon  the  wall  AB.  If 
AK'is  made  perpendicular  to  AB,  and  equal  to  AH,  then 
the  AABK  gives  the  intensity  and  distribution  of  the 

earth-pressure,  or 

E=yAABK. 

The  proof  of  this  construction  is  as  follows  :  Conceive,  in 
Fig.  4,  JD  and  FG  drawn,  then 

A  G  cos  a 


A  nr  —         ~  _  _ 

~  PH~  HG-[AGsma=PGY 

in  which  AP  represents  the  perpendicular  let  fall  from  A 
upon  GH. 


HORIZONTAL  EARTH  SURFACE.  87 

but  AG  :  AF  ::  AF  :  AD=h, 


therefore      AG  =  -^-  =h  tan 

Now 

HG  =  GD  sin  a  —  (A  G  +  AD)  sin  a 

—  h  sin  a  -f-  li  tan2  (45°  -  -  ~)  sin  or; 


tan  AHG  = 


h  tan2  (45°  -  2\  cos  « 


f\  /  fj  \ 

45°  —  -^1  sin  a  —  h  tan2  (45°  -     ^j  sin  a 
2J  \  £/ 

therefore 

tan  AHG  =  ^-^  tan2  ^45°  —  -}  =  cot  a  tan2  (45°  —  *?}. 
sin  a          \  2J  \  2] 


From  Fig.  4,  <GDJ =  <AHG,  <GDJ+<JGD  =  00°, 
and  therefore 

tan  JGD  =  cot  A  HG  —  tan  a  cot  2U5°  —  ^  —  tan  («+tf), 

or  <JGD  is  the  angle  of  the  earth-pressure  to  the  horizon. 
Since,  now,         <AIIG  =  90°  —  a  —  d, 

cos  a  •     a/,e0       o?\         cos  «• 

= 7—    -^r  ^1 G  =  U  tan2   45    —  ^    -  -^T, 

cos  (a  +  6)  \  2]  cos  (« -f  (^) 

and 


88 


THEORY  OF  THE  DETAINING -WALL. 


For  a  vertical  wall  the  construction  becomes  much  sim- 
pler.    Draw,  in   Fig.   5,   AD  —  li  horizontally,  then   DF 

making  the  angle  45°  —  ^  with  AD.     Draw  through  D  and 

& 

circle  with  centre  in  DA  and  continue  it  around  to  K' 


\ 


C    B 


FIG.  5. 

then  the  AABK gives  the  intensity  and  distribution  of  the 
earth-pressure,  while  in  direction  it  is  horizontal. 

Hence  E=yAABK. 

The  proof  is  as  follows  (Fig.  5): 


—  T-~ 

AD 


ft'  tan*  (45°.  -f 

•  —        —  j— 

h 


=  A  tan' (48° -| 


= tan' 45°-       =     . 


HORIZONTAL  EARTH-SURFACE. 


89 


As  a  =  0,  equation  (28)  gives  tan  $  =  0;  .  • .  tf  =  0  and  E 
act  normal  to  the  surface  of  the  wall. 


\ 


_/ 

R  C^    i 


FIG.  6. 

Finally,  in  Fig.  6  is  the  construction  for  loaded  earth- 
surface.  The  point  of  application  of  the  earth-pressure  is 
always  found  by  drawing  through  the  centre  of  gravity  of 
AABK  a  parallel  to  AK  and  producing  it  to  meet  the 
wall.  The  proof  for  this  construction  is  the  same  as  that 
for  Fig.  4. 


90  THEORY  OF  THE  KETAINING-WALL. 


IV. 

EARTH  SURFACE  PARALLEL  TO  SURFACE  OF  REPOSE. 

£=  cp. 
FOR  this  case, 

=  cos*  (<?-<*)  r_y_ _  rcos  (cp  -  a)-l*         l?y     _  m 
eosO  +  d)    2  ~    \_      cos  a      J  2  cos  (a-f-  tf)'  v" 

a  formula  which  holds  good  for  all  values  of  d,  and  which 
for  tf  =  0  or  <p  gives  results  usually  accepted  in  previous 
theories  of  retaining-walls.  In  order  to  find  the  proper 
values  of  d  and  GO,  equations  (16#)  and  (22b)  must  be 
used. 

In  equation  (22b)  replace  sin  (cp  -f-  oo  -{-  a  +  d)  by  sin 
(q> -\-  GO  -\-  a)  cos  #  +  cos  (^  +  GO  -f  a')  sin  tf,  and  making 
s  —cp  it  becomes 

-f-  cos  (cp  -{-  GO)  cos  (<p  -|-  ^  cos  ^ 

-  cos  (cp  -\-  GO)  sin  (9?  +  GO  -j-  a')  cos  d  sin  « 
—  cos  (cp  -\-  GO)  cos  (cp  -\-  GO  -f-  of)  sin  #  sin  a: 

I   +  cos  («  —  <p)  cos  («  -J-  ^)  cos  ^ 

—  cos  (a  —  cp)  fin  (cp -{-  GO -{-  a)  sin  GO  cos 
(^  _  COs  (a  —  cp)  cos  (<f>-\-  GO  -f-  «f)  sin  tf  sin  a? ; 


AKGLE  e  —  ANGLE  cp.  91 

dividing  by  cos  d  and  transposing, 

cos  (ex  —  cp)  cos  (a  -\-  6)  cos  cp 
cos  d 

-j-  cos  (a  —  cp)  sin  (cp  -}-  GO  -j-  a)  sin  GO   {>  = 
+  cos  (cp  -\-  GO)  cos  (<p  +  GO) 

-  cos  (<p  -f  GO)  sin  (^  -|-  GO  -f  a)  sin  ^  J 


SI  T1   O 

-f  cos  (cp  -j-  GO)  cos  (cp  -f  GJ  -j-  a) sin  a 

cos  o 

-  cos  (a  —  cp)  cos  (cp  -j-  GO  -f-<*) -~  sin  GO. 

cos  o 


Since 

COS  (a  -  <f>)  COS  (a  -(-  8)  COS  <f>  _       COS  (a  —  <f>)  COS  <f>  (COS  a  COS  5  —  sin  a  sin  5) 

cos  8  cos  8 

=  —  COS  (a  —  </>)  COS  </>  COS  a  -(-  COS  (a  —  <£)  sin  a COS  0, 

the  above  expression  reduces  to 

tan  8  = 

COS  a  COS(a-<^  COS  <f>—  COS  a  COS  (<^>+a>)  COS  (<f>  +w+  a)  ~  COS(a  —  <f>)  sin  a>  sill((f»-f  w+a) 


sin  a  cos  (a  —  ^)"cos"^"sin  a  cos(</>+  a>)  cos(<J>  +w+a)  -(-  cos(a—  <f>)  sin  to  cos(</>+w  -(-a) 

and  this  equation  fulfils  the  condition  that  the  sum  of  the 
moments  of  G,  E,  and  R  sliall  be  zero. 

If  equation  (1G&)  is  treated  in  a  like  manner,  the  result- 
ing equation  will  fulfil  the  condition  that  the  sum  of  the 
forces  parallel  to  the  surface  of  rupture  shall  equal  zero. 
Making  f  =  cp  in  equation  (16#),  it  reduces  to 


sin  (a  -j-  GO)  cos  (cp  -f-  G?)  cos  (a  -\-  d) 

—  sin  (cp  +  OL  -\-  GO  -j-  $)  cos  (cp  -j-  GO)  cos  (a  —  cp)  =  0? 
3 


92  THEORY  OF  THE  RETAININQ-WALL. 

or 

sin  (a  -|-  GO)  cos  (a  ~\-  d)—  sin  (cp  -\-  GO -\-  a)  cos  («—  <p)  cos  d 
—  cos  (<p  +  GO  -f-  ^)  cos  (o:  —  <p)  sin  d  =  0, 
or 

sin  (a  -}-  <*>)  cos  or  cos  ^      sin  (a  -f-  co)  sin  #  sin  ^ 


cos  6"  cos  d 

\      /          \     cos(<z>-|-Gi>-[-tf)cos(a'—  <»)sin# 
+a)ooB(a-p)—  -^^  v  -=0; 

therefore 

_  cos  a  sin  (a  +  ^  —  sin  (9?  +  &9  +  a)  cos  (a  —  cp) 
~  sin  (a  -f-  G;?)  sin  a  -{-  cos  (f/>  +  GJ  -j-or)  cos  («:  —  cp) 

Setting  both  values  of  tan  6"  equal  to  each  other  and  clear- 
ing of  fractions^  the  following  expression  is  obtained: 

-j-  cos  a  cos  cp  sin  a  sin  (&)  -f-  a)  cos  (a  —  cp) 
-  cos  a  sin  a  sin  (co  -\-  a]  cos  (&?  +  <p)  cos  (&)  +  cp  -{-  a) 

—  sin  Gtfsin  a  sin  (GO  -\-  a)  cos  (a  —  cp)  sin  (<£>  -f-  GO  -f-  tf) 
-)-  cos  a  cos  9?  cos  (a  —  cp)  cos  (<p  +  GO  -f-  ^)  cos  (<?  —  ^>) 

—  cos  a  cos  (<£>  -j-  62?)  cos2  (cp  -{-  GJ  -J-  a)  cos  (<x  —  ^) 

—  sin  co  cos2  (a  —  cp)  sin  (<p  -f  GO  -J-  <*)  cos  (9  +  GO  -|-  a) 

for  the  first  member  of  the  equation,  and 

-j-  cos  a  cos  cp  sin  or  sin  (GO  -f  or)  cos  (or  -f  9?) 
-  sin  <*  cos  a  sin  (cy  -f-  a)  cos  (GJ  -|  --  )  cos  (cp  -{-  co  -\-  a) 
-\-  sin  oo  cos  <*  sin  (a?  -{-  #)  cos  (or  —  cp)  cos  (<p  +  GJ  -(-  ^) 
—  sin  tf  cos  cp  cos2  (^  —  cp)  sin  (<p  -|-  GJ  -j-  a) 


-  sin  G£>  cos2  (a  —  cp)  cos  (<p  +  oo  -\-  a)  sin  (cp  -f-  GL>  -f  « 
for  the  second  member. 


ANGLE  e  =  ANGLE  (p.  93 

The  first  terms,  second  terms,  and  sixth  terms  cancel. 
Divide  the  equation  by  cos  (a  —  cp).  Terms  number  3 
combined  give 

—  sinw  sin(w  +  a)  [sin  a  sin  (<f>  -f  a>  -f-  a)  -f  COS  a  COS  (•£  +  «  + a)], 

which  becomes 

—  sin  GO  sin  (GO  -{-  a)  cos  (cp  -f-  GO). 
Terms  number  5  combined  give 

—  COS  (<t>  +  w)  COS  (<f>  4-  w  -|-  a)  [COS  a  COS  (<£  +  a>  -j-a)  -fsin  a  sin  (<£  +  <•>  +  a)], 

which  becomes 

—  cos  (cp  +  GO  -\-  a)  cos  (cp  -\-  GO)  cos  (cp  +  GO). 
Terms  number  4  combined  give 

-j-  cos  ^  cos  (cr—  ^>)  [cos  a  cos  (p -\- GO -\- a) -\-  sin  a  sin  (<p  +  GO  -f-  <^)]» 
which  becomes 

+  cos  cp  cos  (a  —  cp)  cos  (<p  -f  &?), 

and  hence,  after   dividing  by  cos  (cp  -j-  co),  the  equation 
above  reduces  to 

cos  (a— <p)  cos  cp— cos  (<p-f  co-f-a)  cos  (<p-|-<»)-  sin  (oj-J-cr)  sin  oa^O,  (31) 
and  this  equation  is  fulfilled  for 

Go  =  W°-cp (32) 

In  order  to  find  that  value  of  d  which  satisfies  all  condi- 
tions of  equilibrium,  substitute  the  above  value  of  GO  in  the 

first  expression  for  tan  d  and  obtain  — .       If,  according  to 


94  THEORY  Of1  THE  RETAlNTNG-WALL. 

the  method  for  discussing  indeterminate  fractions,  the 
first  differentials  of  the  numerator  and  denominator  and 
their  ratio  are  found,  and  GJ  made  equal  to  90°  —  <p,  the 
value  of  tan  d  will  be  found. 

The  differential  of  the  numerator  is 


—  cos  acos((^-)-G3)cos(^-f  (a-j-a)—  cos(o:— 
which  equals 


-j-  cos  a' cos  (cp-\-  GO-}-  ct)$\\\  (cp -\- GO] 
+  cos  of  cos  (cp  -f-  GO)  sin  (fp-\-  GO  -j-  <r) 
—  cos  (a  —  cp)  sin  (cp-{-co  -J-  a)  cos  Cc?  | 
—  cos  (a  —  <£>)  sin  GO  cos  (<£>  -f-  &?  -j-  a)  j 

Substituting  for  GO,  90°  —  cp,  this  becomes 

f  -j-  cos  nr  cos  (<p  +  90°—  <p-f-  n')  sin  (  ^-)-  90°  —  cp) 
J  -fcos«  cos  (cp+  90°—  <^)  sin  (^?+  90°  —  ^+  a'} 
I  —  cos  (or—  <p)sin  ((^+90°  —  ^  -j-  a]  cos  (90°—  <p) 

As  the  second  term  reduces  to  zero,  this  becomes 

[cos  a  sin  a  —cos  (a  —  <p)  cos  a  sin  <p  -f-  cos  (a:  —  <p)  cos  <p  sin  a]  c 

or 

Fsin  2fx 

—- cos  (a  —(p)  (cos  a  sin  cp—  cos  9?  sin  a)  Idco, 

tsin  2tf  N~|  , 

— cos  (a  —  cp)  sm  (cp—  a)  \aGD 


or 


=  ANGLE  cp.  95 

or 

2  sin  ^(2cp—  2a  -f  2<x)  cos  ±(2cp  —  2^  — 


2 

which  equals  sin  cpcos(cp  —  2a)  dco. 

Tlie  differential  of  the  denominator  is 
-f-  sin  a  cos  (cp-{-  GO  -\-  a)  sin 


sin  a  cos  (cp  -\-  &))  sin  (<^-|-  GO  -\-  a) 
--  cos  (<v  —q))  cos  (^  -|-  GO  -\-tx}  cos  G!? 
|^  +  cos  (^  ~~  ^)  §in  ^  sin  (<^+  GO'-{-  a) 


Substituting  90°  —  cp  for  GO,  and  this  becomes 

[sin  a  sin  a  -f-  cos  (a  —  93)  sin  a  sin  ^-f"008^  —  <p)cos  q>  cos  a]  6? 

or 

[sin2  a  -\-  cos  («  —  cp)  (sin  9?  sin  a  -j-  cos  <p  cos  a)]dGi), 
or 

[1  —  COS2  a  +  COS  («  —  cp)  COS  (tf  —   ^)]6/Gz7 

cos  2a       I       cos2(o'  —  cp)      1 


or  [1  —  sin  ^>  sin  (cp  —2a)]doj- 

therefore 


tan  <?  = 

1  —  sin  <p  sin  (^?  —  2«) 

To  find  an  expression  for  the  sin  d,  clear  equation  (33) 


96          THEORY  OP  THE  RETAINING -WALL. 

of  fractions  and  deduce  tan  d  —  tan  d  sin  cp  sin  (cp  — 
=  sin  cp  cos  (cp  —  2a).     Multiplying  by  cos  d, 

sin  d  —  sin  d  sin  cp  sin  (<p—  2a)  =  sin  <p  cos  (cp—  2a)  cos  tf, 

or 

sin  6  =  sin  <p  [sin  d  sin  (<p—  2«)  -f-  cos  (cp  —  2<x)  cos  £]; 

therefore 

sin  #  1=  sin  cp  cos  (2<r  —  cp  -\-  6),      .     .     (34) 

from  which  the  results  of  III.  can  be  deduced. 

If  the  earth-surface  is  parallel  to  the  surface  of  repose, 
or  makes  the  angle  cp  with  the  horizontal,  then,  under  the 
assumption  of  a  plane  surface  of  rupture,  d  =  cpou]y  when 
the  wall  is  vertical  (make  a  =  0  in  equation  (33),  then 
tan  6  —  tan  cp;  .  • .  6  =  cp),  and  d  =  0  only  when  the  angle 

of  the  wall  with  the  vertical  a  —  45°  +  -?• 

A 

As  it  is  often  more  convenient  in  determining  the  direc- 
tion of  the  earth-pressure  to  know  the  angle  (a  -j-  d)  of  E 
with  the  horizon,  tan  (a  -|-  6)  may  be  expressed  in  terms  of 
tan  ct  and  tan  tf,  remembering  that 

cos  a  —  sin  cp  sin  (cp  —  a)  —  cos  cp  cos  (cp  —  a), 

and  hence 

,          .,       sin  a  -f  sin  cp  cos  (cp—  oi) 

tan  (a  -f  d)  =  -  —. ±~ — '-.     .  (340) 

cos  cp  cos  (cp  —  a) 

With  reference  to  a  limited  portion  of  wall  which  does 


ANGLE  e  =  ANGLE  (p.  97 

hot  reach  as  far  as  the  surface,  and  with  reference  to 
loaded  earth-surface,  the  same  remarks  hold  good  as  in  III. 

Instead  of  formulae  (20)  and  (33)  or  (34),  the  following 
construction  may  be  used: 

Draw  through  A,  Fig.  7,  a  parallel  to  the  earth -surface. 


FIG.  7. 

and  with  AC  as  a  radius  describe  the  circle  ADG.  Draw 
DF  horizontal  and  GH  parallel  to  AH,  and  then  the 
straight  line HFJ.  Then  the  direction  of  the  earth-pressure 
is  GJ\  and  if  A K  is  made  perpendicular  to  AB  and  equal 
to  HP,  E  —  yAABK,  and  the  triangle  gives  the  distribu- 
tion of  the  pressure.  The  point  of  application  is  found 
by  drawing  through  the  centre  of  gravity  of  the  triangle  a 
perpendicular  to  AB. 


98  THEORY  OF  THE  RETAINING  -WALL. 

The  proof  of  this  construction  is  as  follows  : 
Conceive  HD  drawn,  and  its  intersection  with  GJ  to  be 
at  L.     Then  from  the  notation  of  Fig.  3,  where  f  =  cp, 

FD  =  AD  cos  cp,  HD  =  2  AD  cos  (cp  -  a). 

Since,  no-.v,    <JLD  =  <JHD  -f-  (p  —  a,  by  expressing 
tan  JLD  by  tan  of  JHD  and  cp  —  a,  after  reducing, 


tan  JLD  -      cos 


1  -f-  cos  2(cp  —  a)  —  cos  <£>  cos  (Var  —  (p)' 
or 

sin  c>  cos  (<p  —  %ai) 

tan  JLZ>  =  —  —  .  —  tan  $. 

1  -j-  sin  cp  sin  (<£?  —  x^a) 

Since  HD  is  perpendicular  isAB,  the  earth-pressure  has 
the  direction  GJ.     Further, 


FD  sin  a  sin  #  cos  < 


sin  (or  -|~  <y  —  99)      sin  (a  -\-  d  —  cp] 


AD  —  ——~  -  '.  or.  with  reference  to  the  value  of  FD. 
cos  cp 

cos  (cp  —  a)  sin  aF          ,      . 

AABK  =  —  —  r  —  ,  and   since  from   equation 

sin  (ex  -j-  d  —  cp)  2 

(34)  sin  (a  -\-  d  —  cp)  cos  (cp  —  a)  =  sin  a  cos  (a  -j-  cJ), 


RECAPITULATION  OF  FORMULM 


RECAPITULATION  OF  FORMULAE. 
Inclined  earth-surface,  plane  : 


H    —    |/    I      •    VT-        .       -/    -"'     V-T  -/  QgX 

K   cos  (a  +  ox)  cos  (a  -  £)' 
The  tan  6"  deduced  from  formulae  (226)  and  (161): 

„  sin  (2tf  —  s)  —  A" sin  2(<^  —  f) 

tin  o  =  -= — 

A  —  cos  (2a  —  f)  +  A  cos  2(«  —  e) 

in  which 

COS  £  —     1/COS2  f   —    COS2   CD 


cos2  cp 


_       _ 
L(»  +  1)  cos  ^J  a  cos  (a-  -f  d)' 

Earth-surface  parallel  to  natural  slope  : 


=  _       __ 

L        cos^       J2cos(«+(y)' 

6,9  =  90°-^;       ......     ...     (32). 

tan  («  +  *)=  B^  a  +  8^^008(^-0) 

COS  £>  COS  (^  —  a) 

tan  *  -      sin  <p  coajy-jg^ 
1  -sin  ^sin  (^  -  2«)' 


100  THEORY  OF  THtf  RETAINING-  WALL. 

Horizontal  earth-surface: 


=  45°-|;  .........     (26) 


sin  cp  sin  2a 
tan  <5  =  -  -  ~  —    —r-  ;      ......     (27) 

1  —  sin  cp  cos  2a 


tan 


-—7^ 
tan'    45°  -  f) 


-     •     •     (29) 


sin  (a  +  tf) 

If  a  =  0,  then  d  =  0,  and 


. 

2  cos  (<x  + 

........  (29a) 


If  a  —  (45°  —  ^-)  =  GO,  then  6  =  q>,  and 

2  / 

tan  (45° -f 


sin  [45 


K+f) 


If  the  surface  is  loaded,  substitute  IT  +  7/2  for  A2,  or  con- 
sider li  to  be  the  height  of  the  earth  increased  by  the 
height  of  an  amount  of  earth  weighing  as  much  as  the 
applied  load. 


RECAPITULA T10N  OF  FORM  UL^fl.  101 

NOMENCLATURE. 

Height  of  wall H 

Thickness  at  base b 

Thickness  at  top br 

Batter  in  inches  per  foot  of  Hon  front  face. . .  d 

Weight  per  cubic  foot W 

Total  weight  of  wall G 

Angle  of  repose  of  earth cp 

Angle  made  by  surface  of  rupture  with  vertical  GO 

Weight  of  cubic  foot  of  earth y 

Total  thrust  of  earth  against  wall E 

Angle  made  with  the  horizontal  by  the  surface 

of  the  earth s 

Angle  made  by  rear  face  of  wall  with  the  ver- 
tical  a 

Angle  made  with  normal  by  E. d 

Dist.  of  point  where  the  resultant  pressure  cuts 

the  base  from  the  front  edge  of  the  wall . .   q 
The  resultant  pressure  due  to  E  and  G R 


NOTE. 

FOR  the  translation  of  Prof.  Wey ranch's  paper  the 
writer  is  indebted  to  the  labor  of  Prof.  A.  J.  Du  Bois,  of 
the  Sheffield  Scientific  School,  Yale  College,  who  had 
copies  printed  by  the  electric-pen  process.  However, 
only  the  leading  equations  of  Prof.  Weyrauch  were  given  ; 
hence  a  great  deal  of  labor  has  been  devoted  to  expanding, 
verifying,  and  filling  in  the  intermediate  steps  of  the 
work,  and  this  nucleus  of  the  mathematical  part  alone 
has  grown  to  about  double  the  original  quantity. 

M.  A.  H. 


REFERENCES. 


A  brief  outline  of  the  theories  advanced  by  the  follow- 
ing writers  can  be  found  in  "  Neue  Theorie  des  Erd- 
druckes,"  Dr.  E.  Winkler,  Wien,  1872: 

D' Antony,  Hoffmann,  Poncelet, 

Ande,  Holzhey,  Prony, 

Andoy,  de  Lafont,  Rankine, 

Belidor,  Levi,  Rebhann, 

Blaveau,  deKoszegh  Martony,  Rondelet, 

Bullet,  Maschek,  Saint-Guilhem, 

Considere,  Mayniel,  Saint- Venant, 

Coulomb,  Mohr,  Sallonnier, 

Couplet,  Montlong,  Scheffler, 

Culmann,  Moseley,  Trincaux, 

Frangais,  Navier,  Vauban, 

Gadroy,  Ortmann,  Winkler, 

Gauthey,  v.  Ott,  Woltmann. 

Hagen,  Persy, 

AUDE.  Poussee  des  Terres.     Nouvelles  experiences  sur  la 

poussee  des  terres.     Paris,  1849. 
BAKER-CURIE.  Note  sur  la  brochure  de  M.  B.  Baker  theorie. 

Annales  des  Pouts  et  Chaussees,  pp.  558-592,  1882. 
—  The  actual  lateral  pressure  of  earthwork.     Van  Nos- 
trand's   Magazine,  xxv,  1881;    also   Van   Nostrand's 
Science  Series,  No,  5G. 

103 


104  REFERENCES. 

BOUSSINESQ.  Complement  a  de  precedentes  notes  sur  la 
poussee  des  terres.  *Annales  P.  et  C.,  1884. 

BOUSIN.  Equilibrium  of  pulverulent  bodies.  The  equilib- 
rium of  earth  when  confined  by  a  wall.  fVan  N.,  1881. 

CAIN.  Modification  of  Weyrauch's  Theory.    Van  N.,  1880. 

-  Earth-pressure.    Modification  of  Weyrauch/s  Theory. 
Criticism  of  Baker's  articles.     Van  N.,  1882. 

-  Uniform  cross-section, and  T  abutments:  their  proper 
proportions  and  sizes,  deduced  from  Rankine's  general 
formulas.     Van  N.,  1872. 

-  Practical    designing    of    retaining-walls.      Van    N. 
Science  Series,  No.  3,  1888. 

CHAPERON.  Observations  sur  le  memoire  de  M.  de  Sazilly 
(1851).  Stabilite  et  consolidation  des  talus.  Annales 
P.  et  C.,  1853. 

CONSIDERS.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
C.,  1870. 

COUSINERY.  Determination  graphique  de  1'epaisseur  des 
inurs  de  soutenement.  Annales  P.  et  C.,  1841. 

DE  LAFONT.  Sur  la  poussee  des  terres  et  sur  les  dimensions 
a  donner,  suivant  leurs  profils,  aux  murs  de  soutene- 
ment et  de  reservoirs  d'eau.  Annales  P.  et  C.,  1866. 

DE  SAZILLY.  Sur  les  conditions  d'equilibre  des  massifs  de 
terre,  et  sur  les  revetements  des  talus.  Annales  P.  et 
C.,  1851. 

EDDY.  Retaining-walls  treated  graphically.     Van  N.,  1877. 

FLAMANT.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
C.,  1882. 

Resume  d'articles  publies  par  la  Societe  des  Inge- 

nieures  Civils  de  Londres  sur  la  poussee  des  terres.  An- 
nales P.  et  C.,  1883. 

*  Annales  des  Fonts  et  Chaussees. 
f  Van  Nostrand's  Magazine. 


REFERENCES.  105 

FLAMANT.  Note  sur  la  poussee  des  terres.  Annales  P.  et 
0.,  1872. 

-  Memoire  sur  la  stabilite  de  la  terre  sans  cohesion  par 
W.  J.  Macquorm  Eankine    (Extrait   1856-57).      An- 
nales P.  et  C.,  1874. 

GOBIN.  Determination  precis  de  la  stabilite  des  murs  de 
soutenement  et  de  la  poussee  des  terres.  Annales  P. 
et  C.,  1883. 

GOULD.   Theory  of  J.  Dubosque.     Van  N.,  1883. 

-  Designing.     Van  N.,  1877. 

JACOB.  Practical  designing  of  retaining-walls.  Van  N., 
1873;  also  Van  N.  Science  Series,  No.  3. 

JACQUIEB.  Note  sur  la  determination  graphique  de  la 
poussee  des  terres.  Annales  P.  et  0.,  1882. 

KLEITZ.  Determination  de  la  poussee  des  terres  et  eta- 
blissement  des  murs  de  soutenement.  Annales  P.  et 
C.,  1844. 

LAGREUE.  Note  sur  la  poussee  des  terres  avec  ou  sans  sur- 
charges. Annales  P.  et  C..  1881. 

L'EvEiLLE.  De  1'emploi  des  contre-forts.  Annales  P.  et  C. 
1844. 

LEYGUE.  Sur  les  grands  murs  de  soutenement  de  la  ligne 
de  Mezamet  a  Bedarieux.  Annales  P.  et  C.,  1887. 

-  Nouvelle   recherche  sur  la  poussee   des  terres  et  le 
profil  de  revetement  le  plus  economique.     Annales  P. 
et  C.,  1885. 

MERRIMAN.  On  the  theories  of  the  lateral  pressure  of  sand 
against  retaining  walls.  (School  of  Mines  Quarterly.) 
Engineering  News,  1888. 

—  The  theory  and  calculation  of  earthwork.     Engineer- 
ing News,  1885. 

Theorie  des  Erddruckes  und  der  Futtermauern. 
Wien,  1870  and  1871, 


106  REFERENCES. 

SAINT-GUILHEM.  Sur  la  poussee  des  terres  avec  ou  sans 
surcharge.     Annales  P.  et  C.,  1858. 

ScHEFFLER-FouRNiE.  Traite  de  la  stabilitc  des  construc- 
tions.    Paris,  1864. 

TATE.  Surcharged  and  different  forms  of  retaining- walls. 
Van  N.,  1873;  also  Van  N.  Science  Series,  No.  V 
Also  published  by  E.  &  F.  N.  Spon. 

THORNTON.  Theory.     Van  N.,  1879. 


DIAGRAM  I. 


107 


TABLES. 


Table  I  contains  the  crushing-strengths  and  the  average 
weights  of  stone  likely  to  be  used  in  the  construction  of 
retaining-walls  and  foundations;  also  the  average  weights 
of  different  earths. 

Table  II  contains  the  coefficients  of  friction,  limiting 
angles  of  friction,  and  the  reciprocals  of  the  coefficients  of 
friction  for  various  substances. 

Tables  III,  IV,  and  V  contain  the  values  of  the  coeffi- 
cients [see  equation  (I')]  (B),  (C),  (D)  and  (E),  where 

x      cos  (e  —  a)      /ri.         .   2  (  cos  (e— a] 

(B)  —  — ~      --,     (C)  =  sin2  a,     (D)  —  \ 

cos  a  cos  e  (       cos  e 

,  _,.         .    .          .        cos  (e  —  a) 
and  (E)  =  2  sm  a  sin  e  -  — -. 

cos  e 

The  tables  were  computed  with  a  Thacher  calculating  in- 
strument and  checked  by  means  of  diagrams.  It  is  believed 
that  they  are  correct  to  the  second  place  of  decimals;  an 
error  in  the  third  place  of  decimals  does  not  affect  the  re- 
sults for  practical  purposes. 

Table  VI  contains  the  natural  sines,  cosines  and  tan- 
gents. 

109 


110 


TABLES. 


TABLE  I. 

VALUES  OF    W. 


Name  of  Substance. 

Crushing 
Lds.  in  tons 
per  sq.  ft. 

Average 
weight  in  Ibs. 
per  cu.  ft. 

Alab.ister  

144 

Urick   best  pressed. 

40  to  300 

150 

"       common  hard 

125 

"       soft  inferior     .       

100 

Chalk    

20  to  30 

150 

(  'ement    loose 

49  6  to  102 

Flint 

162 

Feldspar           ....         ....                    . 

160 

Granite     

300  to  1200 

170 

Gneiss  

168 

Greenstone,  trap      

187 

Hornblende   black 

203 

Limestones  and  Marbles   ordinary 

250  to  1000 

j  164.4 

Mortftr  hardened  

I  108 
103 

Quartz    common 

165 

Sandstone 

1  50  to  550 

151 

Shales                      .         ....         .... 

162 

Slate    

400  to  800 

175 

Soapstone  

170 

VALUES   OF 


Name  of  Substance. 

Average 
weight  in  Ibs. 
per  cu.  ft. 

Earth,  common  loam, 
Gravel 

loose  

72  to  80 
82       92 
90       100 
90      106 
90      106 
104      120 
118      129 

shaken 

rammed  moderately  

Sand                     

Sand  nerfectlv  wet  . 

ill 


TABLE  II. 

*  ANGLES  AND    COEFFICIENTS  OF  FRICTION. 


tan  <£. 

* 

tan</> 

Dry  masonry  and  brickwork 
Masonry     and     brickwork 
with  damp  mortar       .... 

O.Gto  0.7 

0  74 

31°  to  35° 
364° 

1.67  to  1.43 
1  35 

Timber  on  stone  

about  0.4 
0.7    toO.3 

22° 

35°    tolfif0 

2.5 
1.43  to  3  33 

Timber  on  timber 

05     "02 

2610  "  lli° 

2  "  5 

Timber  on  metals    

06     "02 

31°     "  11J° 

1  67  "  5 

Metals  on  metals  

0  25  "  0.15 

14°     "    8-T 

4  "  6  67 

Masonry  on  dry  clay  
"          "    moist  clay 

0.51 
0  33 

.  27°.- 
18J° 

1.96 
3 

Earth  on  earth  
Earth   on    earth,  dry  sand, 
clay,  and  mixed  earth.  .  .  . 
Earth  on  earth,  damp  clay  . 
Earth  on  earth,  wet  clay.    . 
Earth  on  earth,  shingle  and 

0.25  to  1.0 

0.38  "0.75 
1.0 
0.31 

0  81 

14°  to  45° 

21°  "  37° 
45° 
17° 

39°  to  48° 

4  to  1 

2.63  "  1.33 
1 
3.23 

1  23  to  0  9 

From  Rankine's  Applied  Mechanics. 


112 


TABLES. 


TABLE  III. 


e 

a  =  5° 

a  =  6° 

a  =  7° 

a  =  6° 

a  =  9° 

(B) 

(B) 

(B) 

(#) 

(B) 

0 

1.004 

1.0U5 

1.007 

1.010 

1.012 

5 

1.012 

1.015 

1.018 

1.022 

1.026 

10 

1.019 

1.024 

1.029 

1.035 

1.040 

15 

1.027 

1.034 

1.041 

.048 

1.055 

20 

1.036 

1.044 

1.052 

.062 

1.071 

25 

1.045 

1.055 

1.065 

.076 

1.088 

30 

1.055 

1.006 

1.079 

.092 

1.105 

35 

1.065 

1.079 

1.094 

1.109 

1.124 

40 

1.078 

1.094 

1.111 

1  .  129 

1.147 

45 

1.093 

1  111 

1.131 

1.152 

1.173 

(C) 

(O) 

(C) 

(C) 

(£) 

0.008 

0.011 

0.015 

0.019 

0.0-^4 

TABLE  IV. 


€ 

a  =5° 

a  =  6° 

a  =  ?° 

a  =  8° 

a  =  9° 

(D) 

(D) 

(D) 

(D) 

(D) 

0 

0.992 

0.989 

0.985 

0.981 

0.976 

5 

1.008 

1.008 

1.006 

1.005 

1.003 

10 

1.023 

1.026 

1.028 

1.030 

1.031 

15 

1.040 

1.046 

1.051 

1.056 

1.060 

20 

1.057 

1.066 

1.075 

1.084 

1.092 

25 

1.075 

1.089 

1.102 

1.114 

1.125 

30 

1.096 

1.113 

1.130 

1.147 

1.163 

35 

1.118 

1.140 

1.164 

1.183 

1.204 

40 

1.144 

1.172 

1.199 

1.226 

1.253 

45 

1.174 

1.208 

1.242 

1.276 

1.309 

TABLE  V. 


6 

a  =  5° 

a  =  6° 

a  =  7° 

a  =  8- 

a  =  9° 

(E) 

(K) 

(E) 

(E) 

(E) 

0 

0 

0 

0 

0 

0 

5 

0.015 

0.018 

0.021 

0.024 

0.027 

10 

0.031 

0.037 

0.043 

0.049 

0.055 

15 

0.046 

0.055 

0.065 

0.074 

0.083 

20 

0.061 

0.074 

0.086 

0.099 

0.112 

25 

0.076 

0.092 

0.108 

0.124 

0.140 

30 

0.091 

0.110 

0.130 

0.149 

0.169 

35 

0.106 

0.128 

0.151 

0.174 

0.197 

40 

0.120 

0.145 

0.172 

0.198 

0.225 

45 

0.134 

0.162 

0.192 

0.222 

0.253 

TABLES. 


113 


TABLE  HI— Continued. 


e 

a=  10° 

a  =llo 

a=  12° 

a=  13° 

a=  14° 

(£) 

(#) 

CB) 

(fi) 

(#) 

0 

1.015 

1.019 

1.022 

1.026 

1.031 

5 

1.031 

1.037 

1.041 

1.047 

1.053 

10 

1.046 

1.055 

1.061 

1.068 

1.076 

15 

1.063 

1.073 

1.081 

:  .090 

1.100 

20 

1.081 

1.092 

1.103 

.112 

1.120 

25 

1.099 

1.112 

1.124 

.136 

1.150 

30 

1.119 

1.135 

1.151 

.163 

1.179 

35 

1.141 

1.159 

1.175 

.195 

1.211 

40 

1.166 

1.186 

1.205 

.225 

1.245 

45 

1.195 

1.218 

1.240 

1.263 

1.288 

(O) 

CO 

(C) 

(C) 

(C) 

0.030 

0.036 

0.043 

0.051 

0.029 

TABLE  IV—  Continued. 


€ 

a=  10° 

a=  11° 

a=  l:>° 

a  =  13° 

a=  14° 

(£) 

(C>) 

CD) 

(*>) 

(£>) 

0 

0.970 

0.964 

0.957 

0.950 

0.943 

5 

.000 

0.997 

0.993 

0.988 

0.983 

10 

.031 

1.031 

.030 

1.028 

1.026 

15 

.064 

1.067 

.069 

1.061 

1.072 

20 

.099 

1.105 

.110 

1.116 

1.121 

25 

.136 

1.147 

.156 

1.165 

1.173 

30 

.178 

1.194 

.204 

1.220 

1.232 

35 

.224 

1.244 

.262 

1.281 

1.300 

40 

.291 

1.304 

1.328 

1.353 

.1.377 

45 

.342 

1.375 

1.407 

1.438 

1.469 

TABLE  V—  Continued. 


e 

a=  10° 

a=  11° 

a=  !•>» 

a  =  13° 

a=  14° 

(#) 

(&') 

(E) 

CS) 

(#) 

0 

0 

0 

0 

0 

0 

5 

0.030 

0.032 

0.036 

0.039 

0.042 

10 

0061 

0.067 

0.073 

0.079 

0.085 

15 

0.093 

0.102 

0.111 

0.119 

0.130 

20 

0.124 

0.137 

0.150 

0.163 

0.175 

25 

0.156 

0.173 

0.189 

0.205 

0.221 

30 

0.188 

0.208 

0.216 

0.248 

0.269 

35 

0.220 

0.244 

0  268 

0.292 

0.316 

40 

0.252 

0.280 

0.308 

0.336 

0.365 

45 

0.284 

0.316 

0.349 

0.382 

0.415 

UHI7BRSIT7 


114 


TABLES. 


TABLE  III— Continued. 


a  =  15° 

a=  16° 

a  =  17° 

a=  18° 

a  =20° 

(#) 

(fi) 

f/0 

(JB) 

OS) 

0 

1.035 

.040 

1.048 

1.051 

1.062 

5 

1.059 

.060 

1.076 

1.081 

1.098 

10 

1.084 

.093 

l.lt)4 

1.112 

1.132 

15 

1.110 

.120 

1.134 

1.138 

1.1(58 

20 

1.135 

.149 

1.165 

1.177 

1.218 

25 

1.165 

.179 

1.197 

1.211 

1.245 

30 

1.195 

.212 

1.233 

1.248 

1.288 

35 

1.229 

.249 

1.272 

1.291 

1.339 

40 

1.268 

.21)1 

1.317 

1.340 

1.389 

45 

1.313 

.338 

1.369 

1.393 

1.451 

(O) 

CO) 

(C') 

CO) 

(CO 

0.067 

0.076 

0  086 

0.095 

0  11? 

TABL E  IV—  Continued. 


e 

a=:  15° 

a=  16° 

a=  17° 

a  =  18° 

a  =  20° 

CM) 

CD) 

(D) 

ID) 

(£>) 

0 

0  933 

0.924 

0.915 

0.905 

0.883 

5 

0.977 

0.971 

0.964 

0  957 

0.940 

10 

1.023 

1.018 

1.016 

1.011 

1  .  000 

15 

1.072 

1.073 

1.071 

1  069 

1.068 

20 

1.124 

1.127 

1.129 

1.181 

1.132 

25 

1.181 

1.188 

1.194 

1.200 

1.208 

30 

1.244 

1.256 

1.266 

1.276 

1.293 

35 

1.316 

1.332 

1.348 

1.363 

1.390 

40 

1.400 

1.422 

1.444 

1.465 

1.505 

45 

1.500 

1.530 

1.559 

1.588 

1.643 

TABLE  V- Continued. 


e 

a  =  15° 

a=  16° 

a=  17° 

a  =  18° 

a=  20° 

CE) 

(0) 

(£) 

(K) 

(#) 

0 

0 

0 

0 

0 

0 

5 

0.045 

0.047 

0.050 

0.053 

0.058 

10 

0.091 

0  097 

0.102 

0.108 

0.119 

15 

0.139 

0.148 

0.157 

0.165 

0.183 

20 

0.188 

0.200 

0.213 

0.225 

0.249 

25 

0.238 

0.254 

0.270 

0.177 

0.318 

30 

0.289 

0  .  309 

0.3>9 

0.349 

0.389 

35 

0  341 

0.365 

0.390 

0.414 

0.463 

40 

0.394 

0.42°, 

0.452 

0.481 

0.539 

45 

0.448 

0.482 

0.516 

0.551 

0.620 

TABLE  VI. 


NATURAL  SINES,  COSINES,  TANGENTS 
AND    COTANGENTS. 


NATURAL  SINES  AND  COSINES. 


Sine 

Tooooo 

.00029 
.00058 
.00087 
.00116 
.00145 
.00175 


One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 
One. 


.01745 
.01774 
.01803 
.01832 

.01862 


.99983 
.99983 
.99982 
.99982 
.99981 
.99980 
.99980 


.99935 
.99934 
.99933 
.99932 
.99931 
.99930 


.01920 
.01949 
.01978 
.02007 


99846' 
99844 
99842 
99841 


00320 

00349 
00378 
00407 
00436 
00465 
00495 
00524 
00553 
00582 

00611 
00640 


02065 
02094 
02123 
02152 
02181 
02211 
02240 
02269 
02298 
02327 


99979 
99978 
99977 
99977 
99976 
99976 
99975 
99974 
99974 


.99888 

.99830 
.99834 
.99833 
.99831 


02414 
02443 
02472 
02501 
02530 


99824 
99822 
99821 
99819 
99817 
99815 
99813 


00727 
00756 
00785 
00814 
00844 
00873 


0-.W.I 

02618 

02647 

02676 
02705 
02734 
02763 
02792 
02821 
02850 


06134 

06163 
06192 
06221 
06250 
06279 
06308 
06337 
06366 
06395 


01018 
01047 
01076 
01105 
01134 
01164 


01193 
01222 
01251 
01280 
01309 
01338 
01367 
01396 
01425 
01454 


99957 
99956 
99955 
99954 
99953 
99952 
90962 


06424 

06453 
06482 
06511 
06540 
06569 


.02967 
.02996 
.03025 
.03054 


99991 
99991 
99991 
99990 


.03112 
.03141 
.03170 
.03199 


06627 
06656 
06685 

06714 
06743 
06773 


.01483 
.01513 
.01542! 
.01571 
.01600 1 
.01629 
.01658 
.01687 
,01716 
01745 


99876 

99875 
99873 
99872 
99870 


.99774 
.99772 
.99770 
.99768 
.99766 
.99764 
.99762 
.99760 
.99758 
.99756 


03257 
03286 
03316 
03345 
03374 
03403 
03432 
03461 


.99947 
.99946 
.99945 
.99944 
.99943 
.99942 


89' 


85° 


NATURAL  SINES  AND  COSINES. 


117 


Sine 


08716 
08745 

08774 


08831 


.08976 
.09005 

.09034 
.09063 
09092 
.09121 
.09150 
.09179 


Cosin 


.99617 
.99614 
.99612 


.99607 
.99604 


.09237 


.09353 
.09382 
.09411 
.09440 


.09498 
.09527 
.09556 
.09585 

.09614 
.09642 
.09671 
.09700 
.09729 
.09758 
.09787 
.09816 
.09845 
.09874 

.09903 
.09932 
.09961 
.09990 
.10019 
.10048 
.10077 
.10106 
.10135 
.10164 

.10192 
.10221 
.10250 
.10279 
.10308 
.10337 
10366 
.10395 
.10424 
.10453 


Cosin 


99580 

99578 
99575 
99572 
99570 
99567 

99564 
99562 
99559 
99556 
99553 
99551 
99548 
99545 
99542 
99540 

99537 
99534 
99531 
99528 
99526 
99523 
99520 
,99517 
,99514 
,99511 


,99506 
,99503 
.99500 
,99497 
,99494 
,99491 
,99488 
,99485 
,99482 

,99479 
.99476 
.99473 
.99470 
.99467 
.99464 
.99461 
.99458 
.99455 
.99452 


Sine 


Sine 
10453 


.10511 
.10540 


.10597 
.10626 
.10655 
.10684 
.10713 
.10742 

.10771 
.10800 


.10858 
.10887 
.10916 
.10945 
.10973 
.11002 
.11031 

.11060 
.11089 
.11118 
.11147 
.11176 
.11205 
.11234 
.11263 
.11291 
.11320 

.11349 
.11378 
.11407 
.11436 
.11465 
.11494 
.11523 
.11552 
.11580 
.11609 

.11638 
.11667 
.11696 
.11725 
.11754 
.11783 
.11812 
.11840 
.11869 
.11898 

.11927 

.11956 
.11985 
.12014 
.12043 
.12071 
.12100 
.12129 
.12158 
.12187 


Cosin 


Cosin 


99443 
99440 
99437 
99434 
99431 
99428 
99424 
99421 

99418 
99415 
99412 
99409 
99406 
99402 


99393 


99377 
99374 
99370 
99367 
99364 


99357 

99354 
99351 
99347 
99344, 
99341 
99337 
99334 
99331 
99327 
99324 


99317 
99314 
99310 
99307 


.99297 
.99293 


.99283 
.99279 
.99276 
.99272 


.99265 
.99262 


^99255 
Sine 


83' 


Sine 


.12187 
.12216 
.12245 
.12274 
.12302 
.12331 
.12360 
.12389 
.12418 
.12447 
.13476 

.12504 
.12533 
.12562 
.12591 
.12620 
.12649 
.12678 
.12706 
.12735 
.12764 

.12793 
.12822 
.12851 

.12880 


,12937 
,12966 
,12995 
,13024 
,13053 

,13081 
,13110 
,13139 
,13168 
,13197 
,13226 
,13254 
,13283 
,13312 
13341 

13370 
,13399 
,13427 
,13450 
,13485 
,13514 
,13543 
,13572 
,13600 
,13629 

,13658 
,13687 
.13716 
.13744 
.13773 
.13802 
.13831 


.13917 


Cosin 


Cosin 


.99248 
.99244 
.99240 
.99237 
.99233 


.99226 
.99222 
.99219 

99215 
99211 
99208 
99204 


99197 
99193 


,99182 

,99178 
,99175 
,99171 
,99167 
,99163 


,99156 
,99152 


,99144 

,99141 

,99137 
,99133 
,99129 
.99125 
,99122 
.99118 
,99114 
.99110 
.99106 

.99102 


99094 
,99091 
,99087 
,99083 
,99079 
,99075 
,99071 
,99067 


.99059 
,99055 
,99051 
,99047 
,99043 
,99039 


.99031 
.99027 
Sine 


82- 


^ 

13917 
13946 
13975 
14004 
14033 
14061 
14090 
14119 
14148 
14177 
14205 

.14234 

.14263 
.14292 
.14320 
.14349 
.14378 
.14407 
.14436 
.14464 
.14493 

.14522 

.14551 

.14580 

.14608 

.1463' 

.14666 

.14695 

.14723 

.1475 

.14781 

.14810 

.14838 
.14867 


.14925 
.14954 
.14982 
.15011 
.15040 
.15069 

.15097 
.15126 
.15155 
.15184 
.15212 
.15241 
.15270 
.1529§ 
.15327 
.15356 

.15385 
.15414 
.15442 
.15471 
.15500 
.15529 
.15557 
.15586 
.15615 
.15643 


Cosin 


Cosin 


.99019 
.99015 
.99011 
.99006 
.99002 
.98998 


.98990 


,98978 
,98973 
,98969 
,98965 
,98961 
,98957 


.98944 


98927 


.98914 

,98910 


,98876 
,98871 
,98867 

,98863 


,98854 

,98849 
,98845 


.98832 


,98818 
.98814 


,98796 
,98791 
,98787 
,98782 
98778 
98773 


Sine 


.15643 
.15672 
.15701 

.15730 
.15758 
.15787 
.15816 
.15845 
.15873 
.15902 
.15931 

.15959 

.15988 
.16017 
.16046 
.16074 
.16103 
.16132 
.16160 
.16189 
.16218 

.16246 
.16275 
.16304 
.16333 
.16361 
.16390 
.16419 
.16447 
.16476 
.16505 

.16533 

.16562 
.16591 
.16620 
.16648 
.16677 
.16706 
.16734 
.16763 
.16792 

.16820 

.16849 
.16878 
.16906 
.16935 
.16964 
.16992 
.17021 
.17050 
.17078 

.17107 
.17136 
.17164 
.17193 
.17222 
.17250 
.17279 
.17308 
.17336 
.17365 


Cosin 


!osin 


.98764 
.98760 
.98755 
.98751 
.98746 
.98741 
.98737 
.98732 
.98728 
.98723 

.98718 
.98714 
.98709 
.98704 
.98700 
.98695 


.98676 

.98671 
.98667 
.98662 
.98657 
.98652 


98629 

98624 
98619 
98614 


.98585 


.98575 
.98570 
.98565 
.98561 
.98556 
.98551 
.98546 
.98541 
.98536 
.98531 

.98526 

.98521 
.98516 
.98511 
.98506 
.98501 
.98496 
.98491 
.98486 
.98481 


Sine 


80' 


118 


NATtRAL  SINES  AND  COSINES. 


34 


37 


10° 


^ 

.17365 
.17393 
.17422 
.17451 
.17479 
.17508 
.17587 
.17565 
.17594 
.17623 
.17651 


17708 
17737 
17766 
17794 
17823 
17852 
17880 
17909 
17937 

17966 

17995 
18023 
18052 
18081 
18109 
18138 
18166 
18195 
18224 


18367 
18395 


18452 

18481 
18509 

18538 
18567 
18595 
18624 
18652 
18681 
18710 
18738 
18767 
18795 


18910 


18967 


19024 
19052 
19081 


Cosin 


Cosin 


.98481 
.98476 
.98471 
.98466 
.98461 
.98455 
.98450 
.98445 


.98435 
.98430 

98425 

98420 
98414 


98404 

98399 
98394 


98378 

98373 
98368 

98362 


98352 

98347 
98341 
98336 


98320 

98315 


98304 


98277 
98272 


98256 
98250 
98245 
98240 
98234 
98229 
98223 
98218 


98190 
98185 
98179 
.98174 
.98168 
.98163 


^790 


11° 


Sine 


.19081 
.1910S 
.19188 
-19167 
19195 
19224 
19252 

19: 

19; 


19423 

19452 
19481 
19509 


19566 
19595 


19652 


19709 
19737 
19766 
19794 
19823 
19851 
19880 
19908 
19937 

19965 
19994 

20022 
20051 
20079 
20108 
20136 
20165 


20279 
20307 


20421 
.20450 
'.20478 
.20507 


.20563 
.20592 
.20620 
: 20849 
'.20677 
".20706 
f.  20734 
>20763 
'.20791 


Cosin 


.98157 
.98153 
.98146 
.98140 
.98135 
.98129 
.98124 
.98118 
.98112 
.98107 

98101 


,98080 

98084 
98079 
98073 


98050 
98044 


93033 
98027 
98021 
98016 
98010 
98004 
97998 
97992 

97987 
97981 
97975 
97969 
97963 
97958 
97952 


97940 
97934 


97922 
97916 
97910 
97905 


97887 
97881 
97875 


97863 
97857 
97851 
'.97845 
: 97839 
'.97833 
'.97827 
.97821 
,97815 


£k>sin  •Sinej 
*  178-r 


Sine 


.20791 


.20848 

.20877 

.20905 

.20933 

.20962 

.20990 

.2101 

.21047 

.21076 

.21104 
.21132 
.21161 
.21189 
.21218 
.21246 
.21275 
.21303 
.21331 


.21388 
.21417 
.21445 
.21474 
.21502 
.21530 
.21559 
.21587 
.21616 
.21644 

.21672 
.21701 

21729 
.21753 

21786 
.21814 
.21843 
.21871 
.21899 


121956 
'.21985 
'.22013 
-.22041 
.22070 
.22098 
.22126 
.22155 
; 22183 
'.22212 


'.22297 
f.  22325 
*  22353 


*22410 
'.22438 
» 22467 
f.  22495 


Cosin 


.97815 
.97809 
.97803 
.97797 
.97791 
.97784 
.97778 
.97772 
.97766 
.97760 
.97754 

.97748 
.97742 
.97735 
.97729 
.97723 
.97717 
.97711 
.97705 


97673 
97667 
97661 
97655 
97648 
97642 
97G36 
97630 

97623 
.97017 
97611 
97604 
9^598 
97592 
.97585 
97579 
97573 
.97566 

.'97560 
.97553 
.97547 
.97541 
.97534 
.97528 
•97521 
.97515 
.97508 
•97502 

?97496 
'.97489 
.97483 
'.97476 
.97470 
'.97463 
197457 
.97450 
; 97444 
197437 


Cosin  •! 
'  '77° 


13C 


Sine 


.22495 
.22523 
.22552 


.22665 
.22693 
.22722 
.22750 
.82778 

.22807 


,22977 
,23005 


.23118 
.23146 
.23175 


.23260 
.23283 
.23316 
.23345 

.23373 

.23401 
.23423 
.23458 
.23486 
.23514 
.23542 
.23571 
.23599 
.23627 

$23656 


.23712 
.23740 
.23769 
'.23797 


'.23853 


.24023 
.24051 
.24079 
.24108 


.24164 
".24192 


Cosin 


Cosin 


.97437 
.97430 
.97424 
.97417 
.97411 
.97404 
,.07398 
.97391 
.97384 
.97378 
.97371 

.97365 
.97358 
.97351 
.97345 
.97338 
.97331 
.97325 
.97318 
.97311 
.97304 

.97298 
.97291 
.97284 
.97278 
.97271 
.97264 
.97257 
.97251 
.97244 
.97237 

97230 
97223 
97217 
97210 
97203 
97196 
,97189 
97182 
97176 
97169 

,97162 
,97155 
,97148 
.97141 
,97134 
.97127 
.97120 
.97113 
.97106 
.97100 

.97093 
,97086 
,97079 
,97072 
,97065 
.97058 
.97051 
,97044 
97037 


Sine 


176- 


14- 


Sine 
.24192 
.24220 
.24249 
.24277 
.84305 
.24333 
.84362 
.84390 
.84418 
.84446 
.24474 

.84503 
.24531 
.24559 
.84587 
.24615 
.84644 
.84672 
.24700 
.24728 
.84756 

.24784 
.24813 

.24841 
.24869 
.24897 
.24925 
.24954 
.24982 
.25010 
.25038 

.85066 
.25094 
.25122 
.25151 
.25179 
.25207 
.25235 
.25263 
.25291 ! 
.25320J 

.25348! 
.25376 
.25404 
.25432, 
.25460 
.25488 
.25516 
.25545 
.25573 
.25601 1 

.25629 
.25657 
.256851 
.25713 
.25741 
.25769 
.25798 
.25826 
.25854 


Cosin 


.97030 
.97023 
.97015 
.97008 
.97001 


96945 


96894 


.96873 


.96851 
.96844 
.96837 


.96815 
.96807 


.96778 
96771 
96764 
.96756 
96749 
.96742 

96734 
.96727 
.96719 
.96712 
.96705 


.96675 
.96667 

.96660 
.96653 
.96645 
.96638 
96630 
.96623 
96615 
96608 
96600 


Cosin  |  Sine 


NATURAL  SINES   AND   COSINES. 


119 


21 


15° 


Sine 


25883 
25910 


25966 


Coein 


.96578 
.96570 
.96562 
.96555 

,260501.96547 
.96540 


16- 


Sine 


26107 
26135  .96524 
.96517 

.26191 1.96509 


.26219 


96502 
96494 


.26359 


.262751.96486 
96479i 
96471 
96463 
.96456 
.26415  .96448 
.26443  .96440 

.26471 

.26500  .96425 
.96417 
,96410 


.26556 
.26584 
.26612 


.26724 

.26752 

.26780 


.87004 

.27032 
.27060 
.27068 
.27116 
.27144 
.27172 
.27200 


.27256 
.27284 

.27812 

.27340 


.27396 
.27424 
.27452 
.27480 
.27508 
.27586 
.27564 


%379 
96371 


96347 
96340 


.96301 


.96277 
.96269 
.96261 


,96214 


Cosin  I 


.96198 
.96190 
.96182 
.96174 
.96166 
.96158 
.96150 
.96142 
.96184 

.98188 

Sine 


74° 


.27564 
.27592 
,27620 
,27648 
,27676 
,27704 
,27731 
,27759 
,27787 
,27815 
,27843 

,27871 

,27899 
,27927 
,27955 
,27983 
,28011 
.28039 
,28067 


Cosin 
.96126 

.96118 
.96110 
.96102: 
.96094 


.96078 
•96070 
.96062 
.96054 


.28123 

.28150 

.28178 


28234 


,28200 
,28318 
28346 

,28374 


,28429 
,28457 
,28485 
,28513 
.28541 
,28569 
,28597 
,28625 
,28652 


,28736 
,28764 


17 
,28875 


,29015 
,29042 

,29070 
,29098 
.29126 
,29154 
.29182 


9G021 
96013 
96005 
95997 


95981 
95972 


95956 
95948 
95940 
,95931 
,95923 
,95915 
,95907 
,95898 
,95890 


.95874 

,95863 
,95857 
,95849 
,95841 
.95832 
.95824 
.95816 
.95807 
.95799 

.95791 

,95782 
.95774 
.95766 
,95757 
.95749 
.95740 
.95732 
.95724 
,95715 

,95707 


,95681 

,95673 


Cosin  Sine 


.95656 
.95647 


.82887 
.82914 
.32942 
.32969 
.82997 
.33024 
.33031 
.83079 
.83106 

.83134 

.33161 
.83189 
.83216 
.83244 
.83271 
.33298 
.83326 
.83353 
.83381 


.94979 
.94970 
.94961 
.94952 
.94943 


95450 
95441 
95433 
93424 
95415 
95407 


.94888 
.94878 
.94869 
.94860 
.94851 
.94842 
.94832 


.94303 
.94293 
.94284 
.94274 
.94264 

.94254 
.94245 


.83408 
.83436 
.3346? 
.83490 
.83518 
.83545 
.3357? 
.83600 
.33627 
.33655 


.94814 
.94805 
.94793 
.94786 
.94777 
.94768 
.94758 
.94740 
.94740 


.95319 
.95310 
.95301 
.952C3 


.94730 

.94721 
.94712 
.94702 
.94693 
.94684 
.94674 
.94665 
.94656 
.9404o 


.95275 

.93260 
.95257 
.95248 
.95240 
.95231 
.95222 
.95213 
.95204 
.95195 

.95186 
.95177 
.95168 
.95159 
.95150 
.95142 
.951&3 
.95124 
.95115 
.95106 
Sine" 


.84120 
.84147 
.84175 
34202 


120 


NATURAL  SINES  AND   COSINES. 


20* 


34202 
34229 
34257 
34284 
34311 
34339 
34366 
34393 
34421 
34448 
84475 


.93337 
.359181.93327 

.93316 
.35973 

.36000 
.36027 


.37757 

.37784 
.37811 


.36135 
.36162 
.36190 
.36217 
.36244 
.36271 
.36298 
.36325 
.36352 
.36379 


.34530 
.34557 
.34584 
.34612 
.34639 


.93232 
.93222 
.93211 
.93201 
.93190 
.93180 
.93169 
.93159 
.93148 


.37865 
.37892 
.37919 
.37946 
.37973 
.37999 


.39474 
.39501 
.39528 
.39555 

.39581 


93799 
93789 
93779 
93769 

93759 

93748 


.34830 
.34857 
.34884 
.34912 
.34939 
.34966 
.34993 


93728 
93718 
93708 
93698 


.39715 
.39741 

.39768 
.39795 

.39822 


.36515 
.36542 
.36569 
.36590 
.36623 
.86650 


93677 
93667 

93657 
93647 
93637 


36677 

36704 
36731 
36758 

36785 


.93031 

.93020 
.93010 
.92999 


350T5 
35102 
35130 
35157 
35184 
35211 


.38349 
.38376 

.384C3 
.38430 
.38450 


.40008 
.40035 
.40062 
.40088 
.40115 
.40141 


.92978 
.92967 
.92956 
.92945 
.92935 


93606 
93596 
93585 
93575 
93565 


.38564 
.38591 
.38617 
.38644 
.38071 
.38G03 


93555 

93544 
93534 
93524 
93514 
93503 
93493 
93483 
93472 
93402 


.40195 
.40221 
.40248 
.40275 
.40301 
.40320 
.40355 
.40381 
.40408 


35347 
35375 
35402 
35429 
35456 
35484 
35511 
35538 
35565 


36975 
37002, 
87029! 
37056 
37083 
37110 
37137 
37164 
37191 


92881 
92870 
92859 
92849 
92838 
92837 


.92816 
.92805 
.92794 
.92784 
.92773 
.92762 
.92751 
.92740 
.92729 
.92718 


35619 
35647 
35674 
35701 
35728 
35755 
35782 
35810 


.93441 
.93431 
.93420 
.93410 
.93400 


.38912 

.38939 
.38966 
.38993 
.39020 
.39046 
.39073 


Cosin 


02060 


.92016 
.92005 
.91994 
.91982 
.91971 
.91959 
.91948 


91925 
91914 
91902 
91891 
91879 
91868, 
918561 
91845 | 
91833 
91822 1 

91810 ! 
91799 
917871 
91775 
91764 
91752 
91741 
91729 
91718 
91706 


.91683 
.91671 
.91660 
.91648 
.91636 
.91625 
.91613 
.91601 
.91590 

.91578 
.91566 
.91555 
.91543 
.91531 
.91519 
.91508 
.91496 
.91484 
.91472 

.40434 '.91461 
.91449 
.91437 
.91425 
.91414 
.91402 


.91378 

.40647  .91366 
.  40674 1. 91355 
Cosin  |  Sine 


_ 

.40674 
.40700 
.40727 
.40753 
.40780 
.40806 


Cosin 
.91355  60 
.91343 !  59 
.91331 !  58 
.91319 
.91307 
.91295 
.91283 


.40860  .91272 
.40886  .91260 
.409131.91248 
.40939  .91236 


91224 


.40992 
.41019 
.41045 
.41072 
.41098 
.41125 
.41151 
.41178 
.41204 

.41231 
.41257 
.41284 
.41310 
.41337 
.41363 
.41390 
.41416 
.41443 
.41469 

.41496 
.41522 
.41549 
.41575 
.41602 
.41628 
.41655 
.41681 
.41707 
.41734 

.41760 
.41787 
.41813 
.41840 
.41866 
.41892 
.41919 
.41945 
.41972 
.41998 

.42024 
.42051 
.42077 
.42104 
.42130 
.42156 
.42183 
.42209 
.42235 
.42262 


.91212 
.91200 
.91188 
.91176 
.91164 
.91152 
.91140 
.91128 
.91116 

.91104 
.91092 
.91080 
.91068 
.91056 
.91044 
.91032 
.91020 
.91008 


,90972 
,90960 


,90911 


,90875 


.90851 


.90814 
.90802- 
.90790 
.90778 
.90766 
.90753 

.90741 

.90729 
.90717 
.90704 
.90692 
.90680 
.90668 
.90655 
.90643 
.90631 


69° 


Cosin  Sine 
65° 


NATURAL  SINES  AND   COSINES. 


25° 

26" 

27° 

28° 

29° 

9 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

$ 

"o 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

i 

.42288 

.90618 

.43863 

.89867 

.45425 

.89087 

.46973 

.88281 

.48506 

.87448 

59 

2 

.42315 

.90606 

.43889 

.89854 

.45451 

.89074 

.46999 

.88267 

.48532 

.87434 

58 

3 

.42341 

.90594 

.43916 

.89841 

.45477 

.89061 

.47024 

.88254 

.48557 

.87420 

57 

4 

.42367 

.90582 

.43942 

.89828 

.45503 

.89048 

.47050 

.88240 

.48583 

.87406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

.42420 

.90557 

.43994 

.89803 

.45554 

.89021 

.47101 

.88213 

.48634 

.87377 

54 

7 

.42446 

.90545 

.44020 

.89790 

.45580 

.89008 

.47127 

.88199 

.48659 

.87363 

53 

8 

.42473 

.90532 

.44046 

.89777 

.45606 

.88995 

.47153 

.88185 

.48684 

.87349 

52 

9 

.42499 

.90520 

.44072 

.89764 

.45632 

.88981 

'.47178 

.88172 

.48710 

.87335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.45658 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

.42552 

.90495 

.'44124 

.89739 

;'45684 

.88955 

T47229 

T88144 

.48761 

.87306 

49 

12 

.42578 

.90483 

.44151 

.89726 

.45710 

.88942 

'.47255 

.88130 

.48786 

.87292 

48 

13 

.42604 

.90470 

.44177 

.89713 

.45736 

.88928 

-.47281 

.88117 

.48811 

.87278 

47 

14 

.42631 

.90458 

.44203 

.89700 

.45762 

.88915 

.47306 

.88103 

.48837 

.87264 

46 

15 

.42657 

.90446 

.44229 

:89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45 

16 

.42683 

.90433 

.44255 

.89674 

.45813 

.88888 

.47358 

.88075 

.48888 

.87235 

44 

17 

.42709 

.90421 

.44281 

.89662 

.45839 

.88875 

.47383 

.88062 

.48913 

.87221 

43 

18 

.42736 

.90403 

.44307 

.89649 

.45865 

.88862 

.47409 

.88048 

.48938 

.87207 

42 

19 

.42762 

.90396 

.44333 

.89636 

.45891 

.88848 

.47434 

.88034 

.48964 

.87193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

„-   -.•*•«• 

.88835 

.47460 

.88020 

.48989 

.87178 

40 

21 

.42815 

.90371 

;44385 

.'89610 

\45942 

.88822 

?47486 

.88006 

.49014 

.87164 

39 

22 

.42841 

.90358 

.44411 

.89597 

.45968 

.88808 

•*.47511 

.87993 

.49040 

.87150 

38 

23 

.42867 

.90346 

.44437 

.89584 

.45994 

.83795 

.47537 

.87979 

.49065 

.87136 

37 

24 

.42894 

.90334 

.44464 

.89571 

.46020 

.88782 

'.47562 

.87965 

.49090 

.87121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.40046 

.88708 

147588 

.87951 

.49116 

.87107 

35 

26 

.42946 

.90309 

.44516 

.89545 

.46072 

.88755 

.47614 

.87937 

.49141 

.87093 

34 

27 

.42972 

.90296 

.44542 

.89532 

.4G097 

.88741 

.47639 

.87923 

.49166 

.87079 

33 

23 

.42999 

.90284 

.44568 

.89519 

.40123 

.88728 

.47665 

.87909 

.49192 

.87064 

32 

29 

.43025 

.90271 

.44594 

.89506 

.46149 

.88715 

.47690 

.87890 

.49217 

.87050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

.43077 

.90246 

f44646 

.89480 

146201 

.'88688 

f47741 

.87868 

.49268 

.87021 

29 

32 

.43104 

.90233 

.44672 

.89467 

.40226 

.88674 

.47767 

.87854 

.49293 

.87007 

28 

33 

.43130 

.90221 

.4-1698 

.89454 

.40252 

.88061 

.47793 

.87840 

.49318 

.86993 

27 

34 

.43156 

.90203 

.44724 

.89441 

.46278 

.88647 

.47818 

.87826 

.49344 

.86978 

26 

35 

.43182 

.90196 

.44750 

.89428 

.40304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

38 

.43209 

.90183 

.44776 

.89415 

.40330 

.88620 

.47869 

.87703 

.49394 

.86949 

24 

37 

.43235 

.90171 

.44802 

.89402 

.46355 

.88007 

.47895 

.87784 

.49419 

.86935 

23 

38 

.43261 

.90158 

.44828 

.89389 

.40381 

.88593 

.47020 

.87770 

.49445 

.86921 

22 

39 

.43287 

.90146 

.44854 

.89376 

.40407 

.88580 

.47940 

.87756 

.49470 

.86906 

21 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 

41 

.43340 

.90120 

.44906 

.89350 

.46458 

.88553 

.47997' 

.87729 

.49521 

.86878 

19 

42 

.43366 

.90108 

.44932 

.89337 

.40484 

.8G539 

.48022 

.87715 

.49546 

.86863 

18 

43 

.43392 

.90095 

.44958 

.89324 

.40510 

.88526 

.48048 

.87701 

.49571 

.86849 

17 

44 

.43418 

.90082 

.44984 

.89311 

.40536 

.88512 

.48073 

.87687 

.49596 

.86834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.40501 

.88409 

.48099  .87673 

.49622 

.86820 

15 

46 

.43471 

.90057 

.45036 

.89285 

.46587 

.88485 

.48124 

.87659 

.49647 

.86805 

14 

47 

.43497 

.90045 

.45062 

.89272 

.46613 

.88472 

.48150 

.87645 

.49672 

.80791 

13 

43 

.43523 

.90032 

.45088 

.89259 

.46639 

.88458 

-.48175 

.87631 

.49697 

.86777 

12 

49 

.43549 

.90019 

.45114 

.89245 

.46064 

.88445 

.48201 

.87617 

.49723 

.86762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

10 

51 

.43602 

.89994 

.45166 

.89219 

.46716 

.88417 

.48252 

.87589 

.49773 

.86733 

9 

52 

.43628 

.89981 

.45192 

.89206 

.40742 

.88404 

.48277 

.87575 

.49798 

.86719 

8 

53 

.43654 

.89968 

.45218 

.89193 

.46767 

.88390 

.48303 

.87561 

.49824 

.86704 

7 

54 

.43680 

.89956 

.45243 

.89180 

.46793 

.88377 

.48328 

.87546 

.49849 

.86090 

6 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5 

56 

.43733 

.89930 

.45295 

.89153 

.46844 

.88349 

.48379 

.87518 

.49899 

.86661 

4 

57 

.43759 

.89918 

.45321 

.89140 

.46870 

.88336 

.48405 

.87504 

.49924 

.86646 

3 

58 

.43785 

.89905 

.45347 

.89127 

.46896 

.88322 

.48430 

.87490 

.49950 

.86632 

2 

59 

.43811 

.89892 

.45373 

.89114 

.46921 

.88308 

.48456 

.87476 

.49975 

.86617 

1 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

86603 

_0 

I 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

i 

64° 

63° 

62°   1 

61°   1 

60° 

NATURAL   SINES   AND   COSINES. 


30* 

31« 

32° 

33° 

34* 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

"o 

.50000 

.86603 

.51504 

85717 

.52992 

84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

.50025 

.86588 

.51529 

85702 

.53017 

84789 

.54488 

.83851 

.55943 

.82887!  59 

2 

.50050 

.86573 

.51554 

85687 

.53041 

84774 

.54513  .83835 

.55968 

.82871  58 

3 

.50076 

.86559 

.51579 

85672 

.53066 

84759 

.54537  .83819 

.55992 

.82855  57 

4 

.50101 

.86544 

.51604 

85657 

.53091 

.84743 

.54561  .8C804 

.56016 

.82839  56 

5 

.50126 

.86530 

.51628 

85642 

.53115 

.84728 

.54586  .83788 

.56040 

.828221  55 

6 

.50151 

.86515 

.51653 

85627 

.53140 

.84712 

.54610 

.83772 

.56064 

.82806  !  54 

7 

.50176 

.86501 

.51678 

85612 

.53164 

.84697  ; 

.54635 

.83756 

.56088 

.82790  53 

8 

.50201 

.86486 

.51703 

85597 

.53189 

.84681 

.54659 

.83740 

.56112 

.82773 

52 

9 

.50227 

.86471 

.51728 

85582 

.53214 

.84666 

.54683  .83724 

.56136 

.82757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650, 

.54708 

.83708 

.56160 

.82741 

50 

11 

.50277 

.86442 

.51778 

.85551 

.53263 

.84635 

.54732 

.83692 

.56184 

.82724 

49 

12 

.50302 

.86427 

.51803 

.85536 

.53283 

.84619 

.54750 

.83676 

.56208 

.82708  48 

13 

.50327 

.86413 

.51828 

85521 

.53312 

.84604  1  .54781 

.83660 

.50232 

.82692  47 

14 

.50352 

.86398 

.51852 

.85506 

.53337 

.84588! 

.54805 

.83045 

.56256 

.82675  1  46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83029 

.56280 

.82659  1  45 

16 

.50403 

.86369 

.51902 

.85476 

.53386 

.84557 

.54854 

.83013 

.56305 

.82643  44 

17 

.50428 

.86354 

.51927 

.85461 

.53411 

.84542 

.54878 

.83597 

.56329 

.82626  43 

18 

.50453 

.86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.50353 

.82610  42 

19 

.50478 

.86325 

.51977 

.85431 

.534GO 

.84511 

.64927 

.83505 

.50377 

.82593  41 

20 

.60503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

.50528 

86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

89 

22 

.50553 

.86281 

.52051 

.85383 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544!  38 

23 

.50578 

.86266 

.52076 

.85370 

.53558 

.84448 

.5502-1 

.83501 

.56473 

.82528 

37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55043 

.83485 

.56497 

.82511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53007 

.84417 

.55072 

.83409 

.56521 

.82495 

35 

26 

.50654 

.86222 

.52151 

.85325 

.53032 

.84402 

.55097 

.83453 

.56545 

.82478 

34 

27 

.50679 

.86207 

.52175 

.85310 

.53056 

.84386 

.55121 

.83437 

.56509 

.82462 

33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.5514-5 

.83421 

.56593 

.82446 

32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55109 

.83405 

.50017 

.82429 

31 

80 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

36 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

82 

.50804 

.86133 

.52293 

.85234 

.53779 

.84308 

.55242 

.83356 

.56689 

.82380 

28 

33 

.50829 

.86119 

.52324 

.85218 

.53804 

.84292 

.55206 

.83340 

.56713 

.82363 

27 

34 

.50854 

.86104 

.52349 

.85203 

.53828 

.84277 

.55291 

.83324 

.56730 

.82347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.85173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86059 

.52423 

.85157 

.53902 

.84230 

.55303 

.83276 

.56808 

.82297 

23 

38 

.50954 

.86045 

.52448 

.85142 

.63926 

.84214 

.55388 

.83200 

.56832 

.82281 

22 

39 

.50979 

.86030 

.52473 

.85127 

.03951 

.84198 

.65412 

.83244 

.56856 

.82264 

21 

40 

.51004 

.86015 

.52498 

.85113 

.53975 

.84182 

.65436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000 

.53522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195  .56928 

.82214  i  18 

43 

.51079 

.85970 

.52572 

.85066 

.54049 

.84135 

.55509 

.83179  .56952 

.82198  17 

44 

.51104 

.85956 

.52597 

.85051 

.54073 

.84120 

.55533 

.83103  .66976 

.82181  16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.65557 

.83147 

.57000 

.82165]  15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.83131 

.57024 

.  82148  j  14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115!  .67047 

.821321  13 

48 

.51204 

.85896 

.52698 

.84989 

.54171 

.84057 

.55630 

.83098  :  !  .57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.64195 

.84041 

.55654 

.83082J  .67095 

.82098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83Q66 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54209 

.83994 

.55726 

.83034 

.67107 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.65750 

.83017 

.57191 

.82032 

7 

54 

.51854 

.85806 

.52844 

.84897 

.54317 

.83902 

.55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54306 

.83930 

.55823 

.82909 

.67262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.829.53 

.57286 

.81905 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.67310 

.81949 

2 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83883 

.65895 

.82920 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

0 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine  I  Cosin 

Sine 

Cosin 

Sine 

I 

59° 

58° 

57°       56° 

55° 

NATURAL  SINES  AND  COSINES. 


35°   | 

SB* 

37* 

SB0 

39* 

9 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

~o 

.57358 

.81915' 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

60 

i 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696 

69 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678 

58 

3 

.57429 

.81865! 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660 

67 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.77641 

56 

5 

.57477 

.818321 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

6 

.57501 

.81815 

.58920 

.80799 

.60321 

.79758 

.61704 

.78694 

.63068 

.77605 

54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586 

53 

8 

.57548 

.81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568 

52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550 

61 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

60 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

'61818 

.78604 

.63180 

.77513 

40 

12 

.57043 

.81714 

.59061 

.80096 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78508 

.63225 

.77476 

47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248 

.77458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16 

.57738 

.81647 

.59154 

.80627 

.60553 

.79583 

.61932 

.78514 

.63293 

.77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.G0576 

.79565 

.61955 

.78496 

.63316 

.77402 

43 

18 

.57786 

.81614 

.59201 

.80593 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384 

42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366 

41 

£0 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

162046 

.78424 

.63406 

.77329 

39 

23 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310 

33 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292 

87 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273 

86 

25 

.57952 

.81406 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

.57976 

.81479 

.59389 

.80455 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236 

34 

27 

.57999 

.81462 

.59412 

.80438 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218 

33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

23 

.58047 

.81423 

.59459 

.80403 

C0853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

SO 

.58070 

.81412 

.59483 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

20 

82 

.58118 

.81378 

.59529 

.80351 

.60922 

.79300 

.62297 

.78225 

.63653 

.77125 

23 

33 

.58141 

.81301 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675 

.77107 

27 

34 

.58165 

.81344 

.59576 

.80316 

.60968 

.79264 

.62342 

.78188 

.63698 

.77088 

2G 

85 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

.58212 

.81310 

.59622 

.80282 

.61015 

.79229 

.62388 

.78152 

.63742 

.77051 

24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211  1  .62411 

.78134 

.63765 

.77033 

23 

38 

.58260 

.81276 

.59669 

.80247 

.61061 

.79193 

.62433 

.78116 

.63787 

.77014 

23 

39 

58283 

.81259 

.59C93 

.80230 

.61084 

.79176 

.62456 

.78098 

.63810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

go 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061 

.63854 

.76959 

19 

42 

.58354 

.81208 

.59763 

.80178 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940 

18 

43 

.58378 

.81191 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921 

17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79C87 

.62570 

.78007 

.63922 

.76903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

.58449 

.81140 

.59856 

.80108 

.61245 

.79051 

.62615 

.77970 

.63966 

.76868 

14 

47 

.58472 

.81123 

.59879 

.80091 

.61268 

.79033 

.62638 

.77952 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.80073 

.61291 

.79016 

.62660 

.77934 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

.58567 

.81055 

.59972 

.80021 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.81038 

.59995 

.80003 

.61383 

.78944 

.62751 

.778G1 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

6 

56 

.58684 

.80970 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.64190 

.76679 

4 

57 

.58708 

.80953 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212 

.76661 

X 

58 

.58731 

.80936 

.60135 

.79899 

.61520 

.78837 

.62887 

.77751 

.64234 

.76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256 

.76623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin, 

Sine 

Cosin 

Sine 

i 

i 

54» 

53° 

52° 

51- 

50° 

124 


NATURAL  SINES  AND  COSINES. 


40° 

41» 

42° 

43" 

44° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

i 

"o 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

.64301 

.76586 

.65628 

.75452 

.66935 

.74295 

.68221 

.73116 

.69487 

.71914 

59 

2 

.64323 

.76567 

.65650 

.75433 

.66956 

.74276 

.68242 

.73096 

.69508 

.71894 

58 

3 

.64346 

.76548 

.65672 

.75414 

.66978 

.74256 

.68264 

.73076 

.69529 

.71873 

57 

4 

.64368 

.76530 

.65694 

.75395 

.66999 

.74237 

.68285 

.73056 

.69549 

.71853 

56 

5 

.64399 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.T1833 

55 

6 

.64412 

.76492 

.65738 

.75356 

.6704S 

.74198 

.68327 

.73016 

.69591 

.71813 

54 

7 

.64435 

.76473 

.65759 

.75337 

.67064 

.74178 

.68349 

.72996 

.69612 

.71792 

53 

8 

.64457 

.76455 

.65781 

.75318 

.67086 

.74159 

.68370 

.72976 

.69633 

.71772 

52 

9 

.64479 

.76433 

.65803 

.75299 

.67107 

.74139 

.68391 

.72957 

.69654 

.71752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

.64524 

.76398 

.65847 

.75261 

.67151 

.74100 

.68434 

.72917 

.69696 

.71711 

49 

13 

.64548 

.7G330 

.65339 

.75241 

.67172 

.74080 

.68455 

.72897 

.69717 

.71091 

48 

13 

.64533 

.76331 

.65891 

.75222 

.67194 

.74061 

.68476 

.72877 

.69737 

.71671 

47 

14 

.64590 

.76342 

.65913 

.75203 

.67215 

.74041 

.68497 

.72857 

.69758 

.71650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

.64635 

.76304 

.65953 

.75165 

.67258 

.74002 

.68539 

.72817 

.69800 

.71610 

44 

17 

.64857 

.76238 

.65078 

75146 

.67280 

.73983 

.68561 

.72797 

.69821 

.71590 

43 

18 

.64679 

.76287 

.63000 

75123 

.67301 

.73963 

.68582 

.72777 

.69842 

.71569 

42 

19 

.64701 

.76248 

.63022 

75107 

.67323 

.73944 

.68603 

.72757 

.69862 

.71549 

41 

20 

.64723 

.70229 

.66044 

75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

.64746 

.76210 

.66066 

75069 

.67366 

.73904 

.68645 

.72717 

.69904 

.71508 

39 

22 

.64768 

.76192 

.63033 

75050 

.  67307 

.73885 

.68GG3 

.72697 

.69925 

.71488 

33 

23 

.64790 

76173 

.63103 

75030 

.67409 

.73805 

.68683 

.72677 

.69946 

.71468 

37 

24 

.64812 

76154 

.63131 

75011 

.67430 

.73846 

.68709 

.72657 

.69966 

.71447 

30 

25 

.64834 

76135 

.68153 

74992 

.67452 

.73823 

.68730 

.72637 

.69987 

.71427 

35 

26 

.64856 

76116 

.68175 

74973 

.67473 

.73803 

.68751 

.72617 

.70008 

.71407 

34 

27 

.64878 

76097 

.66197 

74953 

.67495 

.73787 

.68772 

.72597 

.70029 

.71386 

33 

28 

.64901 

76078 

.66218 

74934 

.67516 

.73767 

.68793 

.72577 

.70049 

.71366 

32 

29 

.64923 

76059 

.68240 

74915 

.67533 

.73747 

.68814 

.72557 

.70070 

.71345 

31 

30 

.64945 

76041 

.66262 

74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

.64967 

76022 

.66284 

74876 

67580 

.73708 

.68857 

.72517 

.70112 

.71305 

29 

32 

.64989 

76003 

.63303 

74857 

67602 

.73683 

.68878 

.72497 

70132 

.71284 

23 

33 

.65011 

75984 

.68327 

74833 

67C23 

.73609 

.68899 

.72477 

.70153 

.71264 

27 

34 

.65033 

75965 

.68349 

74818 

67645 

.73643 

.68920 

.72457 

.70174 

.71243 

26 

.65055 

7594S 

.68371 

74799 

67006 

.73623 

.68941 

.72437 

.70195 

.71223 

35 

38 

.65077 

75927 

.63393 

74780 

67G33 

.73010 

.68902 

.72417 

.7C215 

.71203 

24 

37 

.65100 

75903 

.66414 

74760 

67709 

73590 

.68983 

.72397 

.70236 

.71182 

23 

S3 

.65122 

75889 

.66433 

74741 

67730 

73570 

.69004 

.72377 

.70257 

.71162 

22 

39 

.65144 

75870 

.63453 

74722 

67752 

73551 

.69025 

.72357 

.70277 

.71141 

21 

40 

.65166 

75851 

.66480 

74703 

67773 

73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

.65188 

75832 

.66501 

74683 

67795 

.73511 

.69067 

.72317 

.70319 

.71100 

19 

42 

.65210 

75813 

.63523 

74664 

678fG 

.73491 

.69038 

.72297 

.70339 

.71080 

18 

43 

.65232 

75794 

.68545 

74644 

67837 

.73472 

.69109 

.72277 

.70360 

.71059 

17 

44 

.65254 

75775 

.66563 

74625 

67859 

.73452 

.69130 

.72257 

.70381 

.71039 

16 

45 

.65276 

75756 

.66588 

74608 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

.65298 

75738 

.66610 

74588 

.67901 

.73413 

.69172 

.72216 

.70422 

.70998 

14 

47 

.65320 

75719 

.66632 

74567 

.67923 

.73393 

.69193 

.72196 

.70443 

.70978 

13 

48 

.65342 

75700 

.66653 

74548 

.67944 

.73373 

.69214 

.72176 

.70463 

.70957 

12 

49 

.65364 

75680 

.66675 

74523 

.67965 

.73353 

.69235 

.72156 

.70484 

.70937 

11 

50 

.65386 

75661 

.66697 

74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

.65408 

.75642 

.66718 

74489 

.68008 

.73314 

.69277 

.72116 

.70525 

.70896 

9 

52 

.65430 

.75623 

.66740 

74470 

.68029 

.73294 

.69298 

.72095 

.70546 

.70875 

8 

53 

.65452 

.75604 

.66762 

74451 

.68051 

.73274 

.69319 

.72075 

.70567 

.70855 

7 

54 

.65474 

.75585 

.66783 

.74431 

.68072 

.73254 

.69340 

.72055 

.70587 

.70834 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608 

70813 

5 

56 

.65518 

.75547 

.66827 

.74392 

.68115 

.73215 

.69382 

.72015 

.70628 

70793 

4 

57 

.65540 

.75528 

.66848 

.74373 

.68136 

.73195 

.69403 

.71995 

.70649 

70772 

3 

58 

.65562 

.75509 

.66870 

.74353 

.68157 

.73175 

.69424 

.71974 

.70670 

.70752 

2 

59 

.65584 

.75490 

.66891 

.74334 

.68179 

.73155 

.69445 

.71954 

.70690 

.70731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

49° 

48° 

47° 

40°   i 

45° 

| 

NATURAL  TANGENTS  AND  COTANGENTS. 


125 


0° 

1° 

2° 

8« 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755 

59 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

0 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03096 

27.0566 

.05445 

18.3G55 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2G77 

52 

0 

.00262 

381.971 

.02007 

49.8157 

.03754 

26.63G7 

.05503 

18.1708 

51 

10 

.00291 

&43.T74 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

236.478 

.02095 

47.7395 

.03342 

26.0307 

.05591 

17.88G3 

48 

13 

.00378 

234.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015 

46 

15 

.00436 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106 

45 

1(5 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205 

44 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314 

43 

18 

.00524 

100.984 

.02269 

44.0661 

.04016 

24.8978 

.05766 

17.3432 

42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

80 

.00583 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

82 

.00040 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990 

38 

23 

,OOGG9 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

21 

.  00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

x).> 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

•„';; 

.00756 

132.219 

.02502 

39.9G55 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

2s 

.00315 

122.774 

.02560 

39.05G8 

.04308 

23.2137 

.06058 

16.5075 

32 

20 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

CO 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

81 

.00002 

110.892 

.02648 

37.7683 

'.04395 

22.7519 

.06145 

16.2722 

29 

32 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952 

28 

88 

.009GO 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

84 

.00989 

101.107 

.02735 

36.5G27 

.04483 

22.3081 

.06233 

16.0435 

26 

85 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

87 

.01076 

93.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

::s 

.01105 

90.4633 

.02851 

35.0005 

.04599 

21.7426 

.06350 

15.7483 

22 

.7;l 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762 

21 

40 

.01164 

85.9398 

.02910 

34.3678 

.04658 

21.4704 

.06408 

15.6048 

20 

11 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

12 

.01222 

81.8470 

.029G8 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

ta 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

41 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.2571 

15 

•16 

.C1338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

17 

.01367 

73.1390 

.03114 

32.1181 

.048G2 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

•ID 

.01425 

70.1533 

.03172 

31.5284 

.04920 

20.32.53 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

7 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.03463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

GO 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

t 

89° 

88° 

87*     - 

86° 

126 


NATURAL  TANGENTS   AND   COTANGENTS. 


4° 

5° 

6° 

70 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

GO 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

58 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12307 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.19426 

8.04756 

K 

6 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

60 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12603 

7.93438 

40 

12 

.07344 

13.6174 

.09101 

10.9882 

.1CSG3 

9.20516 

.12CS3 

7.91582 

48 

18 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12602 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.1CC22 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

1G 

.07461 

13.4039 

.09218 

10.8483 

.10901 

9.10(k6 

.12751 

7.84242 

44 

IT 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

48 

IS 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.057C9 

.12810 

7.80022 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11C70 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.00983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

30 

22 

.07636 

13.0958 

.00394 

10.6450 

.11158 

8.90227 

.12929 

7.73480 

3!< 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

21 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.C9957 

86 

25 

.07724 

-12.9469 

.09482 

10.54G2 

.11246 

8.89185 

.13017 

7.C8208 

85 

20 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.CG4G6 

34 

27 

.07782 

12.8496 

.C9541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

88 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13100 

7.63005 

82 

29 

.07841 

12.7536 

.oceoo 

10.4172 

.11364 

8.79904 

.13136 

7.01287 

81 

oO 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

.13165 

7.59575 

30 

81 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

20 

:J2 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.EGGO 

.00717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806 

26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132 

25 

30 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465 

24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.02078 

.13372 

7.47806 

23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154 

22 

39 

.08134 

12.2946 

.00893 

10.1080 

.11659 

8.57718 

.13432 

7.44509 

31 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688     8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

8.53402 

.13491 

7.41240 

10 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

40 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38G25 

.13698 

7.30018 

12 

41) 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12i01 

8.26355 

.13876 

7.206G1 

G 

55 

.08002 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7.19125 

6 

50 

.08032 

11.5853 

.10393 

9.62205 

.12100 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

5!) 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

GO 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

85° 

84° 

83° 

82° 

NATURAL  TANGENTS   AND   COTANGENTS. 


127 


8° 

< 

9° 

1 

0° 

1 

1- 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~0 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

c 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

s 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

56 

5 

.1420S 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

G 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

54 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19640 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.982G8 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

50 

11 

.14381 

6.95385 

'.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.10190 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.10283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89388 

.16286 

6.14023 

.18033 

5.53007 

.19891 

5.02734 

45 

1C 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.8G874 

'.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

;  16376 

6.10604 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49356 

.20012 

4.99695 

41 

20 

.14648 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

116465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.46648 

.20103 

4.97438 

38 

23 

.14737 

6.73564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04051 

.18353 

5.44857 

.20164 

4.95945 

36 

23 

.14796 

6.75838 

.16585 

6.02902 

.18384 

5.43966 

.20194 

4.95201 

35 

23 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14356 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

as 

28 

.14386 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

32 

29 

.14915 

6.70450 

.16704 

5.98646 

.18504 

5.40429 

.20315 

4.92249 

31 

SO 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

81 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

'.20376 

.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93365 

.18654 

5.36070 

.20466 

.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

.87882 

25 

30 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

.87162 

24 

37 

.15153 

6.50021 

.16944 

5.90191 

.18745 

5.33487 

.20557 

.86444 

23 

30 

.15183 

6.53627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

.85727 

22 

CO 

.15213 

6.57339 

.17094 

5.88114 

.18805 

5.31778 

.20618 

.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.258SO 

.20830 

.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

6.23391 

.20921 

.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

.77286 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

.73851 

5 

56 

.15719 

6.30165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

.73170 

4 

57 

.15749 

6.31961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

.72490 

3 

53 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

.70463 

0 

f 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

8 

!•             1 

8 

0° 

7 

9° 

7 

B«          . 

128 


NATURAL  TANGENTS   AND   COTANGENTS. 


12° 

13° 

14°           ] 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

f 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.23117 

4.32573 

.24904 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.G9121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.C8452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4.67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

G 

.21438 

4.6G458 

.23271 

4.29724 

.25118 

3.98117 

.2G982 

3.70616 

54 

7 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

8 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27044 

3.69761 

52 

9 

.21529 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

13 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27109 

3.680G1 

48 

13 

.21G51 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

14 

.21G82 

4.61219 

.23516 

4.25239 

.253G6 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.272G3 

3.66796 

45 

1C 

.21743 

4.59927 

.23578 

4.24132 

.25428 

3.93271 

.27294 

3.66376 

44 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

10 

.21004 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

42 

19 

.21834 

4.58001 

.23G70 

4.22481 

.25521 

3.91639 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25552 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

S.90890 

.27451 

3.64289 

39 

23 

.21925 

4.5G091 

.23762 

4.20842 

.25014 

3.90417 

.27482 

3.63874 

38 

23 

.21956 

4.55458 

.23793 

4.20298 

.25045 

3.89945 

.27513 

3.63461 

37 

21 

.21986 

4.54826 

.23823 

4.19756 

.25076 

3.89474 

.27545 

3.63048 

36 

23 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

23 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27007 

3  62224 

34 

27 

.22078 

4.52941 

.23916 

4.18137 

.25769 

3.88068 

.27038 

3.61814 

33 

23 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27733 

3  60588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

S3 

.22231 

4  49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

28 

83 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

24 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

33 

.22322 

4.47986 

.24163 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

30 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58100 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

22 

29 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

19 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

•13 

.22567 

4.43134 

.24408 

4.09699 

.26266 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

•3.54968 

16 

43 

.22028 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

40 

.22058 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22089 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.282G6 

3.53785 

13 

<18 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

8.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

Ha 

.22872 

4  37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

51 

.22903 

4.3G623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

6 

£"> 

.22934 

4  30040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

DC 

.22904 

4.35459 

.24809 

4.03076 

.26670 

3.74950 

.28549 

3.50279 

4 

57 

.22995 

4  34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

..24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.26764 

3.73640 

.28643 

3.49125 

1 

CO 

.23087 

4.33148 

.24933 

4.01078 

.2G795 

3.73205 

.28675 

3.48741 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

77° 

76° 

75° 

74" 

NATURAL  TANGENTS  AND  COTANGENT?. 


129 


16° 

17° 

18° 

19° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.28675 

3.48741 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

60 

1 

.28706 

3.48359 

.30605 

3.26745 

.32524 

3.07464 

.34465 

2.90147 

59 

2 

.28738 

3.47977 

.30637 

3.26406 

.32556 

3.07160 

.34498 

2.89873 

58 

3 

.28769 

3.47596 

.30669 

3.26067 

.32588 

3.06857 

.34530 

2.89600 

57 

4 

.28800 

3.47216 

.30700 

3.25729 

.32621 

3.06554 

.34563 

2.89327 

56 

5 

.28832 

3.46837 

.30732 

3.25392 

.32653 

3.06252 

.34596 

2.89055 

55 

6 

.28864 

3.46458 

.30764 

3.25055 

.32685 

3.05950 

.34628 

2.88783 

54 

7 

.28895 

3.46080 

.30796 

3.24719 

.32717 

3.05649 

.34661 

2.88511 

53 

8 

.28927 

3  45703 

.30828 

3.24383 

.32749 

3.05349 

.34693 

2.88240 

52 

9 

.28958 

3.45327 

.30800 

3.24049 

.32782 

3.05049 

.34726 

2.87970 

51 

10 

.28990 

3.44951 

.30891 

3.23714 

.32814 

3.04749 

.34758 

2.87700 

50 

11 

.29021 

3.44576 

.30923 

3.23381 

.32846 

3.04450 

.34791 

2.87430 

49 

12 

.29053 

3.44202 

.30955 

3.23048 

.32878 

3.04152 

.34824 

2.87161 

48 

1-3 

.29084 

3.43829 

.30987 

3.22715 

.32911 

3.03854 

.34856 

2.86892 

47 

14 

.29116 

3.43456 

.31019 

3.22384 

.32943 

3.03556 

.34889 

2.86624 

46 

15 

.29147 

3.43084 

.31051 

3.22053 

.32975 

3.032GO 

.34922 

2.86356 

45 

16 

.29179 

3.42713 

.31083 

3.21722 

.33007 

3.02963 

.34954 

2.86089 

44 

17 

.29210 

3.42343 

.31115 

3.21392 

.33040 

3.02GG7 

.34987 

2.85822 

43 

18 

.29242 

3.41973 

.31147 

3.21063 

.33072 

3.02372 

.35020 

2.85555 

42 

19 

.29274 

3.41604 

.31178 

3.20734 

.33104 

3.02077 

.35C52 

2.85289 

41 

20 

.29305 

3.41236 

.31210 

3.20406 

.33136 

3.01783 

.35085 

2.85023 

40 

21 

.29337 

3.40869 

.31242 

3.20079 

.33169 

3.01489 

.35118 

8.84758 

29 

22 

.29368 

3.40502 

.31274 

3.19752 

.33201 

3.011C6 

.351CO 

2.84494 

38 

23 

.29400 

3.40136 

.31208 

3.19426 

.33233 

3.00C03 

.35183 

2.84229 

37 

24 

.29432 

3.39771 

.31338 

3.10100 

.33266 

3.00611 

.35216 

2.83965 

36 

25 

.29463 

3.39406 

.31370 

3.18775 

.33298 

3.00319 

.35248 

2.83702 

35 

2G 

.29495 

3.39042 

.314C2 

3.10451 

.33330 

3.00028 

.35231 

2.83439 

34 

27 

.29526 

3.38679 

.31434 

3.10127 

.33363 

2.99738 

.35314 

2.83176 

33 

28 

.29558 

3.38317 

.31466 

3.17804. 

.33305 

2.99447 

.35346 

2.82914 

32 

29 

.29590 

3.37955 

.31498 

8.17401 

.33427 

2.99158 

.35379 

2.82653 

31 

30 

.29621 

3.37594 

.31530 

3.17159 

.33460 

2.98868 

.35412 

2.82391 

30 

31 

.29653 

3.37234 

.31562 

3.16838 

.33492 

2.98580 

.35445 

2.82130 

29 

33 

.29685 

3.36375 

.31594 

3.16517 

.33524 

2.98292 

.35477 

2.81070 

28 

33 

.29716 

3.3G516 

.31626 

3.16197 

.33557 

2.98004 

.35510 

2.81610 

27 

34 

.29748 

3.30158 

.31658 

3.15877 

.33589 

2.97717 

.35543 

2.81350 

26 

85 

.29780 

3.35800 

.31690 

3.15558 

.33621 

2.97430 

.35576 

2.81091 

25 

36 

.29811 

3.35443 

.31722 

3.15240 

.33654 

2.97144 

.35603 

2.80833 

24 

37 

.29843 

3.35087 

.31754 

3.14922 

.33686 

2.96850 

.35641 

2.80574 

23 

38 

.29875 

3.34732 

.31786 

3.14605 

.33718 

2.96573 

.35674 

2.80316 

22 

39 

.29906 

3.34377 

.31018 

3.14288 

.33751 

2.96288 

.35707 

2.80059 

21 

40 

.29938 

3.34023 

.31850 

3.13972 

.33783 

2.96004 

.35740 

2.79802 

20 

41 

.29970 

3.33670 

.31882 

3.13656 

.33816 

2.95721 

.35772 

2.79545 

19 

42 

.30001 

3.33317 

.31914 

3.13341 

.33848 

2.95437 

.35805 

2.79289 

18 

43 

.30033 

3.32965 

.31946 

3.13027 

.33881 

2.95155 

.35838 

2.79033 

17 

44 

.30065 

3.32614 

.31978 

3.12713 

.33913 

2.94872 

.35871 

2.78778 

16 

45 

.30097 

3.32264 

.32010 

3.12400 

.33945 

2.94591 

.35904 

2.78523 

15 

^6 

.30128 

3.31914 

.32043 

3.12087 

.33978 

2.94309 

.35937 

fc.  78269 

14 

4, 

.30160 

3.31565 

.32074 

3.11775 

.34010 

2.94028 

.85969 

2.78014 

13 

48 

.30192 

3.31216 

.32106 

3.11464 

.34043 

2.93748 

.36002 

2.77761 

12 

49 

.30224 

3.30868 

.32139 

3.11153 

.34075 

2.934C8 

.36035 

2.77507 

11 

50 

.30255 

3.30521 

.32171 

3.10843 

.34108 

2.93189 

.36068 

2.77254 

10 

51 

.30287 

3.30174 

.32203 

3.10532 

.34140 

2.92910 

.36101 

2.77002 

9 

52 

.30319 

3.29829 

.32235 

3.10223 

.34173 

2.92632 

.36134 

2.76750 

8 

53 

.30351 

3.29483 

.32267 

3.09914 

.34205 

2.92354 

.36167 

2.76498 

7 

54 

.30382 

3.:  29139 

.32299 

3.09606 

.34238 

2.92076 

.36199 

2.76247 

6 

55 

.30414 

3.28795 

.32331 

3.09298 

.34270 

2.91799 

.36232 

2.75996 

5 

56 

.30446 

3.28452 

.32363 

3.08991 

.34303 

2.91523 

.36265 

2.75746 

4 

57 

.30478 

3.28109 

.32396 

3.08685 

.34335 

2.91246 

.36298 

2.75496 

3 

58 

.30509 

3.27767 

.32428 

3.08379 

.34368 

2.90971 

.36331 

2.75246 

2 

59 

.30541 

3.27426 

.32400 

3.08073 

.34400 

2.90696 

.36364 

2.74997 

1 

GO 

.30573 

3.27085 

.32492 

3.07768 

.34433 

2.90421 

.36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

73° 

72° 

71° 

70° 

130          NATURAL  TANGENTS   AND    COTANGENTS. 


80- 

21* 

22° 

23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

1 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

2 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

3 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.4G888 

.42551 

2.35015 

57 

4 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46082 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

6 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

7" 

.36628 

2.73017 

.38620 

2.58932 

.40643 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40074 

2.45860 

.42722 

2.34069 

52 

9 

.30694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.83881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

.40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71793 

.38787 

2.57815 

.40809 

2.45043 

.42800 

2.33317 

48 

13 

.30826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.30859 

2.71305 

.38854 

2.57371 

.40877 

2.44G36 

.42929 

2.32943 

46 

15 

.30892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

16 

.33925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.30958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

10 

.30991 

2.70335 

.88988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39023 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

20 

.37057 

2.69853 

.89055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.87090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.CD371 

.39123 

2.55608 

.41149 

2.43019 

.43205 

2.31456    38 

23 

.37157 

2.C3131 

.39156 

2.55389 

.41183 

2.42819 

.43239 

2.31271  |37 

24 

.37190 

2.63392 

.89190 

2.55170 

.41217 

2.42018 

.43274 

2.31086    36 

25 

.3?223 

2.63G53 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902 

3T> 

20 

.37256 

2.C3414 

.39257 

2.54734 

.412C5 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.C3175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534    33 

23 

.37323 

2.67037 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351    32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41307 

2.41620 

.43447 

2.30167    31 

30 

.37388 

2.67462 

.89391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.63989 

.33458 

2.53432 

.41490 

2.41025 

.43550 

2.29619    28 

33 

.37488 

2.GG752 

.33492 

2.53217 

.41524 

2.40827 

.43585 

2.29437    27 

34 

.37521 

2.63516 

.39526 

2.53001 

.41553 

2.40029 

.43620 

2  29254 

26 

35 

.37554 

2.GG281 

.39559 

2.53708 

.41592 

2.40432 

.43654 

2..  29073 

25 

36 

.37588 

2.60046 

.39593 

2.52571 

.416C6 

2.40235 

.43689 

2.28891 

24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41600 

2.40038 

.43724 

2.28710 

23 

33 

.37654 

2.65576 

.39060 

2.52142 

.41694 

2.89841 

.43758 

2.28528 

22 

39 

.37687 

2.C5342 

.39094 

2.51929 

.41728 

2.39045 

.43793 

2.28348 

21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.89761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.61G42 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2  27806 

18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41805 

2.38803 

.43932 

2.27626 

17 

44 

.37853 

2.64177 

.39802 

2.50804 

41899 

2.38008 

.43966 

2.27447 

16 

45 

.37887 

2.63045 

.39898 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

46 

.37920 

2.63714 

.39930 

2.50440 

.41908 

2.38279 

.44036 

2.27088 

14 

47 

.37953 

2.63483 

.33963 

2.50229 

.42003 

2.38084 

.44071 

2.20909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.33020 

2.63021 

.40031 

2.49807 

.42070 

2.37097 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.87504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.87311 

.44210 

2.26196 

9 

52 

33120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.33186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25603 

6 

55 

.38220 

2.61646 

.40234 

2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

248340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

38320 

2  60963 

40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2  24780 

1 

GO 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

j— 
'•/ 

Cotang 

Tang, 

Cotang 

Tang 

Cotang 

Tang 

Cotang     Tang 

/ 

69° 

>  68° 

67'          II          66° 

NATURAL  TANGENTS   AND   COTANGENTS. 


131 


2 

40 

2 

5° 

2 

6' 

2 

7o 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

0 

.44523 

2.24604 

.46631 

2.14451 

.48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

1.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

1.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

1.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

1.95698 

56 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

1.95557 

55 

G 

.44732 

2.23553 

.46843 

2.13477 

.48989 

2.04125 

.51173 

1.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

1.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

1.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

1.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12833 

.49134 

2.03526 

.51319 

1.94858 

50 

11 

144907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

1.94718 

49 

12 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

1.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

1.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

1.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

1.94162 

45 

18 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

1.94023 

44 

ir 

.45117 

2.21647 

.47234 

2.11711 

.49337 

2.02403 

.51577 

1.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

1.93746 

42 

19 

.45187 

2.21304 

.47C05 

2.11392 

.49459 

2.02187 

.51651 

1.93608 

41 

90 

.45222 

2.21133 

.47341 

2.11233 

.49495 

2.02039 

.51688 

1.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

1.93332 

39 

29 

.45292 

2.20790 

.47412 

2.10916 

.495C8 

2.01743 

.51761 

1.93195 

38 

28 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

1.93057 

37 

•J4 

.45362 

•2.20449 

.47483 

*.  10600 

.49640 

2.01449 

.51835 

1.92920 

36 

JJ5 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

1.92782 

35 

26 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

1.92645 

34 

2T 

.45407 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

1.92508 

33 

28 

.45502 

2.19709 

.47626 

2.09909 

.49786 

2.00862 

.51983 

1.92371 

32 

29 

.45538 

2.19599 

.47062 

2.09811 

.49822 

2.00715 

.52020 

1.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09854 

.49858 

2.00569 

.52057 

1.92093 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

1.91962 

29 

3-2 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

1.91826 

28 

83 

.45678 

2.18923 

.47005 

2.09184 

.49967 

2.00131 

.52168 

1.91690 

27 

84 

.45713 

2.18755 

.47840 

2.09028 

.50004 

.99986 

.52205 

1.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

.99841 

.52242 

1.91418 

25 

3G 

.45784 

2.18419 

.47912 

2.08716 

.50076 

.99695 

.52279 

1.91282 

24 

37 

.45819 

2.10251 

.47948 

2.08560 

.50113 

.£9550 

.52316 

1.91147 

23 

88 

.45854 

2.18084 

.47984 

2.08405 

.50149 

.C3406 

.52353 

1.91012 

22 

39 

.45889 

2.17916 

.43019 

2.03250 

.50185 

.C9261 

.52390 

1.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

.93116 

.52427 

1.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.43127 

2.07785 

.50295 

.93828 

.52501 

1.90472 

18 

43 

.40030 

2.17249 

.48163 

2.07630 

.50331 

.93684 

.52538 

1.90337 

17 

41 

.4G065 

2.17083 

.48198 

2.07476 

.50368 

.98540 

.52575 

1.90203 

16 

45 

.40101 

2.10917 

.48234 

2.07321 

.50404 

.08396 

.52613 

1.90069 

15 

46 

.40136 

.2.10751 

.48270 

2.07167 

.50441 

.88253 

.52650 

1.89935 

14 

47 

.40171 

2.10585 

.48306 

2.07014 

.50477 

.93110 

.52687 

1.8S801 

13 

48 

.46206 

2.10420 

.48342 

2.00860 

.50514 

.97966 

.52724 

1.89667 

12 

49 

.46242 

2.10255 

.43378 

2.06706 

.50550 

.97823 

.52761 

1.89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

.97538 

.52836 

1.8926B 

9 

52 

.40348 

2.157GO 

.43486 

2.00247 

.50660 

.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.43521 

2.00094 

.50696 

.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.43557 

2.05942 

.50733 

.97111 

.52947 

1.88807 

6 

68 

.40454 

2.15268 

.48593 

2.05790 

.50769 

.96969 

.52985 

1.88734 

5 

B6 

.40489 

2.15104 

.48629 

2.05637 

.50806 

.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

.96685 

.53059 

1.88469 

3 

58 

.40560 

2.14777 

.48701 

2.05333 

.50879 

.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

.96261 

.53171 

1.88073 

J) 

§ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

6 

5° 

6 

4° 

6 

3° 

6 

z-       1 

132          NATURAL  TANGENTS   AND   COTANGENTS. 


f 

28° 

29° 

30° 

31° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

j 

"o 

.53171 

1.88073 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80281 

.57774 

1.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.55507 

1.80158 

.57813 

1.72973 

.60165 

1.66209 

58 

3 

.53283 

1.87677 

.55545 

1.80034 

.57851 

1.72857 

.60205 

1.66099 

57 

4 

.53320 

1.87546 

.55583 

1.79911 

.57890 

1  .72741 

.60245 

1.65990 

56 

i 

.53358 

1.87415 

.55621 

1.79788 

.57929 

1.72625 

.60284 

1.65881 

55 

6 

.53395 

1.87283 

.55659 

1.79665 

.57968 

1.72509 

.60324 

1.65772 

54 

7 

.53432 

1.87152 

.55697 

1.79542 

.58007 

1.72393 

.60364 

1.65663 

53 

6 

.53470 

1.87021 

.55736 

1.79419 

.58046 

1.72278 

.60403 

1.65554 

52 

9 

.53507 

1.86891 

.55774 

1.79296 

.58085 

1.72163 

.60443 

1.65445 

51 

10 

.53545 

1.86760 

.55813 

1.79174 

.58124 

1.72047 

.60483 

1.65337 

50 

11 

.53582 

1.86630 

.55850 

1.79051 

.58162 

1.71932 

.60522 

1.65228 

49 

12 

.53620 

1.86499 

.55888 

1.78929 

.58201 

1.71817 

.60562 

1.65120 

48 

13 

.53657 

1.86369 

.55926 

1.78807 

.58240 

1.71702 

.60602 

1.65011 

47 

14 

.53694 

1.86239 

.55964 

1,.  78685 

.58279 

1.71588 

.60642 

1.64903 

46 

15 

.53732 

1.86109 

.56003 

1.78563 

.58318 

1.71473 

.60681 

1.64795 

45 

16 

.53769 

1.85979 

.56041 

1.78441 

.58357 

1.71358 

.60721 

1.64687 

44 

17 

.53807 

1.85850 

.56079 

1.78319 

.58396 

1.71244 

.60761 

1.64579 

43 

18 

.53844 

1.85720 

.56117 

1.78198 

.58435 

1.71129 

.60801 

1.64471 

42 

19 

.53882 

1.85591 

.56156 

1.78077 

.58474 

1.71015 

.60841 

1.64363 

41 

20 

.53920 

1.85462 

.56194 

1.77955 

.58513 

1.70901 

.60881 

1.64256 

40 

21 

.53957 

1.85333 

.56232 

1.77834 

.58552 

1.70787 

.60921 

1.64148 

39 

22 

.53995 

1.85204 

.56270 

1.77713 

.58591 

1.70673 

.60960 

1.64041 

38 

23 

.54032 

1.85075 

.56309 

1.77592 

.58631 

1.70560 

.61000 

1.63934 

37 

24 

.54070 

1.84946 

.56347 

1.77471 

.58670 

1.70446 

.61040 

1.63826 

36 

25 

.54107 

1.84818 

.56385 

1.77351 

.58709 

1.70332 

.61080 

1.63719 

35 

26 

.54145 

1.84689 

.56424 

1.77230 

.58748 

1.70219 

.61120 

1.63612 

34 

27 

.54183 

1.84561 

.56462 

1.77110 

.58787 

1.70106 

.61160 

1.63505 

33 

28 

.54220 

1.84433 

.56501 

1.76990 

.58826 

1.69992 

.61200 

1.63398 

32 

29 

.54258 

1.84305 

.5C539 

1.76869 

.588C5 

1.69879 

.61240 

1.63292 

31 

30 

.54296 

1.84177 

.56577 

1.76749 

.58905 

1.69766 

.61280 

1.63185 

30 

31 

.54333 

1.84049 

.56616 

1.76629 

.58944 

1.69653 

.61320 

1.63079 

29 

32 

.54371 

1.83922 

.50654 

1.76510 

.58983 

1.69541 

.61360 

1.62972 

28 

33 

.54409 

1.83794 

.56693 

1.76390 

.59022 

1.69428 

.61400 

1.62866 

27 

34 

.54446 

1.83667 

.56731 

1.76271 

.59061 

1.69316 

.61440 

1.62760 

26 

35 

.54484 

1.83540 

.56769 

1.76151 

.59101 

1.69203 

.61480 

1.62654 

25 

36 

.54522 

1.83413 

.56808 

1.76032 

.59140 

1.69091 

.61520 

1.62548 

24 

37 

.54560 

1.83286 

.56846 

1.75913 

.59179 

1.68979 

.61561 

1.62442 

23 

38 

.54597 

1.83159 

.56885 

1.75794 

.59218 

1.68866 

.61601 

1.62336 

22 

39 

.54635 

1.83033 

.56923 

1.75675 

.59258 

1.68754 

.61641 

1.62230 

21 

40 

.54673 

1.82906 

.56962 

1.75556 

.59297 

1.68643 

.61681 

1.62125 

20 

41 

.54711 

1.82780 

.57000 

1.75437 

.59336 

1.68531 

.61721 

1.62019 

19 

42 

.51748 

1.82654 

.57039 

1.75319 

.59376 

1.68419 

.61761 

1.61914 

18 

43 

.54786 

1.82528 

.57078 

1.75200 

.59415 

1.68308 

.61801 

1.61808 

17 

44 

.5^964 

1.82402 

.57116 

1.75082 

.59454 

1.68196 

.61842 

1.61703 

16 

45 

.54862 

1.82276 

.57155 

1.74964 

.59494 

1.68085 

.61882 

1.61598 

15 

46 

.54900 

1.82150 

.57193 

1.74846 

.59533 

1.67974 

.61922. 

1.61493 

14 

47 

.54938 

1.82025 

.57232 

1.74728 

.59573 

1.67863 

.61962 

1.61388 

13 

48 

.54975 

1.81899 

.57271 

1.74610 

.59612 

1.67752 

.62003 

1.61283 

12 

49 

.55013 

1.81774 

.57309 

1.74492 

.59651 

1.67641 

.62043 

1.61179 

11 

50 

.55051 

1.81649 

.57348 

1.74375 

.59691 

1.67530 

.62083 

1.61074 

10 

51 

.55089 

1.81524 

.57386 

1.74257 

.59730 

1.67419 

.62124 

1.60970 

9 

52 

.55127 

1.81399 

.57425 

1.74140 

.59770 

1.67309 

.62164 

1.60865 

8 

53 

.55165 

1.81274 

.57464 

1.74022 

.59809 

1.67198 

.62204 

1.60761 

7 

54 

.55203 

1.81150 

.57503 

1.73905 

.59849 

1.67088 

.62245 

1.60657 

6 

55 

.55241 

1.81025 

.57541 

1.73788 

.59888 

1.66978 

.62285 

1.60553 

5 

56 

.55279 

1.80901 

.57580 

1.73671 

.59928 

1.66867 

.62325 

1.60449 

4 

57 

.55317 

1.80777 

.57619 

1.73555 

.59967 

1.66757 

.62366 

1.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

1.66647 

.62406 

1.60241 

2 

59 

.55393 

1.80529 

.57696 

1.73321 

.60046 

1.66538 

.62446 

1.60137 

60 

.55431 

1.80405 

.57735 

1.73205 

.60086 

1.66428 

.62487 

1.60033 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

i           61- 

60° 

59» 

58°           i 

NATURAL  TANGENTS   AND   COTANGENTS. 


133 


32" 

83° 

34°           | 

35' 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~o 

62487 

1.  600*3 

.64941 

1.53986 

.67451 

1.48256  i 

.70021 

1.42815 

60 

i 

62527 

1.59930 

.64983 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

2 

62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

62689 

1.59517 

.65148 

1.53497 

.67663 

1.47792 

.70238 

1.42374 

55 

6 

62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

g 

62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

62893 

1.59003 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

12 

62973 

1.58797 

.65438 

1.52816 

.67900 

1.47146 

.70542 

1.41759 

48 

13 

63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

14 

63055 

1.58593 

65521 

1.52623 

.68045 

1.46962 

.70629 

1.41584 

46 

15 

63095 

1.58490 

.65563 

1.52525 

.68088 

1.4G870 

.70673 

1.41497 

45 

16 

.63136 

1.58388 

!  65604 

1.52429 

.68130 

1.46778 

.70717 

1.41409 

44 

17 

.63177 

1.58286 

.65646 

1.52333 

.68173 

1.46G86 

.707GO 

1.41322 

43 

18 

.63217 

1.581&4 

.65G88 

1.52235 

.68215 

1.46595 

.70804 

1.41235 

42 

19 

.63258 

1.58083 

.65729 

1.E2139 

.68258 

1.4G503 

.70S48 

1.41148 

41 

20 

.63299 

1.57981 

.65771 

1.52043 

.68301 

1.46411 

.70891 

1.41061 

40 

21 

.63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70935 

1.40974 

39 

22 

.63380 

1.577T3 

.65854 

1.51850 

.68386 

1.4G229 

.70979 

1.40887 

38 

23 

.63421 

1.57076 

.65893 

1.51754 

.68429 

1.46137 

.71023 

1.40800 

37 

24 

.63462 

1.57575 

.65938 

1.51058 

.68471 

1.4G046 

.71066 

1.40714 

3G 

25 

.63503 

1.57474 

.65980 

1.61563 

.68514 

1.45955 

.71110 

1.40G27 

35 

26 

.63544 

1.57372 

.66021 

1.51406 

.68557 

1.458G4 

.71154 

1.40540 

S4 

27 

.63584 

1.57271 

.66003 

1.51370 

.68600 

1.45773 

.71108 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.68642 

1.45683 

.71242 

1.40367 

32 

29 

.63666 

1.57009 

.66147 

1.51179 

.68685 

1.45593 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.68728 

1.45501 

.71329 

1.40195 

30 

31 

.63748 

1.56868 

.66230 

1.50988 

.68771 

1.45410 

.71373 

1.40109 

29 

12 

.63789 

1.56767 

.66272 

1.50893 

.68814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.68857 

1.45229 

.71461 

1.39936 

27 

34 

.63871 

1.56566 

.66356 

1.50702 

.G8900 

1.45139 

.71505 

1.39850 

2G 

35 

.63912 

1.56466 

.66398 

1.60607 

.68943 

1.45049 

.71549 

1.39764 

25 

36 

.63953 

1.56366 

.66440 

1.50513 

.68985 

1.44958 

.71593 

1.39679 

24 

37 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50323 

.C9071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.66566 

1.50228 

.C9114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.W158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.C9243 

1.44418 

.71857 

1.89165 

18 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

1G 

45 

.64322 

1.55467 

.66818 

1.49661 

.69372 

1.14149 

.71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55263 

.66902 

1.49473 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69503 

1.43881 

.72122 

1.38G53 

12 

4( 

.64487 

1.55071 

.66CCS 

1.49284 

.69545 

1.43793 

.72167 

1.385C8 

11 

& 

.64528 

1.54972 

.67023 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

5 

.64560 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.38399 

9 

5 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

53 

.64652 

1  .54675 

.67115 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

54 

.64693 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

5o 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43358 

.72432 

1.88060 

5 

5 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

ti 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37801 

3 

5* 

.64858 

1.54183 

.67CGe 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

5 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

(X 

.64941 

1.53986 

.67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

( 

Cotang 

Tang 

i  Cotang 

Tang 

Cotang 

Tang 

,  Cotang 

Tang 

57' 

56° 

55°           II           54° 

134 


NATURAL  TANGENTS  AND  COTANGENTS. 


31 

5° 

3 

7° 

3 

B" 

3 

90            1 

Tangf 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

1.27994 

.80978 

.23490 

00 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

1.27917 

.81027 

.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

1.27841 

.81075 

.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

1.27764 

.81123 

.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

1.27688 

.81171 

.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

1.27611 

.81220 

.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

1.27535 

.81268 

.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

1.27458 

.81316 

.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

1.27382 

.81364 

.22904 

52 

9 

.73055 

1.36883 

.75767 

1.31984 

.78551 

1.27306 

.81413 

.22*31 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

1.27230 

.81461 

.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

1.27153 

.81510 

.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78692 

1.27077 

.81558 

.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31666 

.78739 

1.27001 

.81606 

.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

1.26925 

.81655 

.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

1.26849 

.81703 

.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

1.26774 

.81752 

.22321 

44 

17 

.73413 

1.36217 

.76134 

1.31348 

.78928 

1.26698 

.81800 

.22249 

43 

18 

.73457 

1.36134 

.76180 

1.31269 

.78975 

1.26622 

.81849 

.22176 

42 

19 

.73502 

1.36051 

.76226 

1.31190 

.79022 

1.26546 

.81898 

.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

1.26471 

.81946 

.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

1.26395 

.81995 

.21959 

39 

22 

.73637 

1.35802 

.76364 

1.30952 

.79164 

1.26319 

.82044 

.21886 

38 

23 

.73681 

1.35719 

.76410 

1.30873 

.79212 

1.26244 

.82092 

.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

1.26169 

.82141 

.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79306 

1.26093 

.82190 

.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

1.26018 

.82238 

.21598 

34 

27 

.73861 

1.35389 

.76594 

1.30558 

.79401 

1.25943 

.82287 

.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

1.25867 

.82336 

.21454 

32 

29 

.73951 

1.35224 

.70686 

1.30401 

.79496 

1.25792 

.82385 

.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

1.25717 

.82434 

.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

1.25642 

.82483 

.21238 

29 

32 

.74086. 

1.34978 

.76825 

1.30166 

.79639 

1.25567 

.82531 

.21166 

28 

33 

.74131 

1.34896 

.76871 

1.30037 

.79686 

1.25492 

.82580 

.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

1.25417 

.82629 

.21023 

26 

35 

.74221 

1.34732 

.76964 

1.29931 

.79781 

1.25343 

.82678 

.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

1.25268 

.82727 

.20879 

24 

37 

.74312 

1.34568 

.77057 

1.29775 

.79877 

1.25193 

.82776 

.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

1.25118 

.82825 

.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

1.25044 

.82874 

.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

1.24969 

.82923 

.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

1.24895 

.82972 

.20522 

19 

42 

.74538 

1.34160 

.77289 

1.29385 

.80115 

1.24820 

.83022 

.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

1.24746 

.83071 

.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

1.24672 

.83120 

.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

1.24597 

.83169 

.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

1.24523 

.83218 

.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

1.24449 

.83268 

.20095 

13 

48 

.74810 

1.33C73 

.77568 

1.28919 

.80402 

1.24375 

.83317 

.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

1.24301 

.83366 

.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

1.24227 

.83415 

.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

1.24153 

.83465 

.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

1.24079 

.83514 

.19WO 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

1.24005 

.82564 

.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

1.23931 

.83613 

.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

1.23858 

.83662 

.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

1.23784 

.83712 

.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

1.23710 

.83761 

.19387 

8 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

1.23637 

.83811 

.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

,80930 

1.23563 

.83860 

.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

1.23490 

.83910 

.19175 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

t 

I 

3° 

C 

2" 

1 

!• 

j 

0* 

NATURAL  TANGENTS  AND  COTANGENTS.         135 


40° 


Tang 


.83910 


.84009 


.84108 
.84158 


.84258 
.84307 
.84357 
.84407 

.84457 
.84507 
.84556 
.84606 
.84656 
.84706 
.84756 
.84806 


.84056 


.85057 
.85107 
.85157 

.85207 
.85257 


.85358 

.85408 

.85458 


.85660 


.85761 
.85811 

.85862 


.85963 
.86014 


.86115 


.86216 


.86419 
.86470 


.86572 


.86674 
.86725 
.86776 


.86878 


Cotang 


Cotang 


.19175 
.19105 
.19035 
.18964 


.18824 
.18754 


.18614 
.18544 
.18474 

.18404 


.18264 
.18194 
.18125 
.18055 
.17986 
.17916 
.17846 
.17777 

.17708 
.17638 


.17500 
.17430 
.17361 
.17292 


.17154 
.17085 

.17016 
.16947 
.16878 
.16809 
.16741 
.16672 


.16535 
.16466 


.16329 


.16192 
.16124 
.16056 
.15987 
.15919 
.15851 
.15783 
.15715 

.15647 
.15579 
.15511 
.15443 
.15375 
.15308 
.15240 
.15172 
.15104 
.15037 
Tang 


49° 


41° 


Tang 


.87031 

.87082 
.87133 

.87184 


.87287 
.87338 
.87389 
.87441 

.87492 
.87543 
.87595 
.87646 


.87749 
.87801 


.87904 
.87955 


.88007 


.88214 

.88265 


.88421 


.88524 
.88576 


.88784 


.89410 
.89463 
.89515 


.89777 


.89935 

.89988 

.90040 

Cotang 


Cotang 


1.15037 
1.14969 


1485* 
14767 


.14565 
.14498 
.14430 
.14363 

.14296 
.14229 
.14162 
.14095 


.13761 


.13627 
.13561 
.13494 
.13428 
.13361 


.13096 
.13029 

.12963 
.12897 
.12831 
.12765 


.12633 
.12567 
.12501 
.12435 


.12303 


.12172 
.12106 
.12041 
.11975 
.11909 
.11844 
11778 
.11713 

.11648 
.11582 
.11517 
.11452 
.11387 
.11321 
.11256 
.11191 
.11126 
.11061 
Tang 


Tang 

.90040 


.90146 
.90199 
.90251 
.90304 
.90357 
.90410 


.90516 


.90674 
.90727 
.90781 


.90940 


.91046 
.91099 

.91153 

.91206 
.91259 
.91313 


.91419 
.91473 


.91580 


.91687 
.91740 
.91794 
.91847 
.91901 
.91955 


.92062 
.92116 
.92170 

.92224 
.92277 
.92331 


.92493 
.92547 
.92601 
.92655 
.92709 


.92817 
.92872 


.93143 

.93197 

.93252 

Cotang 


Cotang 


1.11061 
1 


.10802 
.10737 
.10672 
.10607 
.10543 
.10478 
.10414 

.10349 

.10285 


.10156 
.10091 
.10027 


.09770 
.09706 


.09578 
.09514 
.09450 


.09195 
.09131 

.09067 


.08876 
.C3813 
.C8749 


.03559 
.084% 


.CS369 
.08306 
.08243 
.C3179 
.08116 
.08053 
.07090 
.07927 
.07864 

.07801 
.07738 
.07676 
.07613 
.07550 
.07487 
.07425 


.07209 
.07237 
Tang 


48° 


47° 


43° 


Tang 


.93415 


.93742 
.93797 


.94016 
.94071 
.94125 

.94180 


.94345 

.94400 
.94455 
.94510 
.94565 
.94620 
.94676 
.94731 
.94786 
.94841 


.94952 
.95007 


.95118 
.95173 
.95229 


.95451 

.95506 
.95562 
.95618 
.95673 
.95729 
.95785 
.95841 
.95897 
.95952 
.96008 

.96064 
.96120 
.96176 


.96400 
.96457 
.96513 
.96569 


Cotang 


Cotang 


.07237 
.07174 
.07112 
.07049 


.06738 
.06676 
.06613 

.06551 

.06489 
.06427 


.06179 
.06117 
.06056 
.05994 


.05870 
.05809 
.05747 
.05685 
.05624 
.05562 
.05501 
.05439 
.05378 

.05317 
.05255 
.05194 
.05133 
.05072 
.05010 
.04949 


.04766 

.04705 

.04644 
.04583 
.04522 
.04461 
.04401 
.04310 
.04279 
.04218 
.04158 

.04097 
.04036 
.03976 
.03915 
.03855 
.03794 
.03734 


03553 
Tang 


46° 


136 


NATURAL  TANGENTS  AND   COTANGENTS. 


440 

440 

44. 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

.03553 

60 

20 

.97700 

.02355 

40 

40 

.98843 

.01170 

20 

1 

.96625 

.03493 

59 

21 

,97756 

.02295 

'39 

41 

.98901 

.01112 

19 

2 

.96681 

.03433 

58 

22 

.97813 

.02236 

38 

42 

.98958 

.01053 

18 

8 

.96738 

.03372 

57 

23 

.97870 

.02176 

87 

43 

.99016 

.00994 

17 

4 

.9679-1 

.03312 

56 

24 

.97927 

.02117 

36 

44 

.99073 

.00935 

16 

5 

.96850 

.03252 

55 

25 

.97984 

.02057 

35 

45 

.99131 

.00876 

15 

6 

.96907 

.03192 

54 

26 

.98041 

.01998 

34 

46 

.99189 

.00818 

14 

.96963 

.0313* 

53 

27 

.98098 

.01939 

33 

47 

.99247 

.00759 

13 

8 

.97020 

.03072 

52 

28 

.98155 

.01879 

32 

48 

.99304 

.00701 

12 

9 

.97076 

.03012 

51 

29 

.98213 

.01820 

81 

49 

.99362 

.00642 

11 

10 

.97133 

.02952 

50 

30 

.98270 

.01761 

30 

50 

.99420 

.00583 

10 

11 

.97189 

.02892 

49 

31 

.98327 

.01702 

20 

51 

.99478 

.00525 

9 

12 

.97246 

.02832 

48 

32 

.98384 

.01642 

28 

52 

.99536 

.00467 

8 

13 

.97302 

.02772 

47 

.98441 

.01583 

27 

53 

.99594 

.00408 

7 

14 

.97359 

.02713 

46 

34 

.98499 

.01524 

26 

54 

.99652 

.00350 

6 

15 

.97416 

.02653 

45 

.35 

.98556 

.01465 

25 

55 

.99710 

.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

.01406 

24 

56 

.99768 

.00233 

4 

17 

.97529 

1.02533 

43 

37 

.98671 

.01347 

23 

57 

.99826 

.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

.01288 

22 

58 

.99884 

.00116 

2 

19 
90 

.97643 
.97700 

1.02414 
1.02355 

41 
40 

39 
40 

.98786 
.98843 

.01229 
.01170 

21 
20 

59 
60 

.99942 
1.00000 

.00058 
.00000 

1 

0 

Cotang 

Tang 

' 

Cotang 

Tang 

t 

Cotang 

Tang 

/ 

45° 

45- 

45- 

/&£'   OF  THE^^^. 

{TJ'HIVEKSITTJ 

X^JFogl^: 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


7 


11 

APR  16   1946 


30m  6/14 


YB   11067 


